diff --git a/src/client/conn/http2.rs b/src/client/conn/http2.rs
index c7938e303b..94c56e6e49 100644
--- a/src/client/conn/http2.rs
+++ b/src/client/conn/http2.rs
@@ -230,7 +230,7 @@ where
     B::Data: Send,
     E: Unpin,
     B::Error: Into<Box<dyn Error + Send + Sync>>,
-    E: Http2ClientConnExec<B, T> + 'static + Send + Sync + Unpin,
+    E: Http2ClientConnExec<B, T> + Unpin,
 {
     type Output = crate::Result<()>;
 
@@ -457,3 +457,219 @@ where
         }
     }
 }
+
+#[cfg(test)]
+mod tests {
+
+    #[tokio::test]
+    #[ignore] // only compilation is checked
+    async fn send_sync_executor_of_non_send_futures() {
+        #[derive(Clone)]
+        struct LocalTokioExecutor;
+
+        impl<F> crate::rt::Executor<F> for LocalTokioExecutor
+        where
+            F: std::future::Future + 'static, // not requiring `Send`
+        {
+            fn execute(&self, fut: F) {
+                // This will spawn into the currently running `LocalSet`.
+                tokio::task::spawn_local(fut);
+            }
+        }
+
+        #[allow(unused)]
+        async fn run(io: impl crate::rt::Read + crate::rt::Write + Unpin + 'static) {
+            let (_sender, conn) = crate::client::conn::http2::handshake::<
+                _,
+                _,
+                http_body_util::Empty<bytes::Bytes>,
+            >(LocalTokioExecutor, io)
+            .await
+            .unwrap();
+
+            tokio::task::spawn_local(async move {
+                conn.await.unwrap();
+            });
+        }
+    }
+
+    #[tokio::test]
+    #[ignore] // only compilation is checked
+    async fn not_send_not_sync_executor_of_not_send_futures() {
+        #[derive(Clone)]
+        struct LocalTokioExecutor {
+            _x: std::marker::PhantomData<std::rc::Rc<()>>,
+        }
+
+        impl<F> crate::rt::Executor<F> for LocalTokioExecutor
+        where
+            F: std::future::Future + 'static, // not requiring `Send`
+        {
+            fn execute(&self, fut: F) {
+                // This will spawn into the currently running `LocalSet`.
+                tokio::task::spawn_local(fut);
+            }
+        }
+
+        #[allow(unused)]
+        async fn run(io: impl crate::rt::Read + crate::rt::Write + Unpin + 'static) {
+            let (_sender, conn) =
+                crate::client::conn::http2::handshake::<_, _, http_body_util::Empty<bytes::Bytes>>(
+                    LocalTokioExecutor {
+                        _x: Default::default(),
+                    },
+                    io,
+                )
+                .await
+                .unwrap();
+
+            tokio::task::spawn_local(async move {
+                conn.await.unwrap();
+            });
+        }
+    }
+
+    #[tokio::test]
+    #[ignore] // only compilation is checked
+    async fn send_not_sync_executor_of_not_send_futures() {
+        #[derive(Clone)]
+        struct LocalTokioExecutor {
+            _x: std::marker::PhantomData<std::cell::Cell<()>>,
+        }
+
+        impl<F> crate::rt::Executor<F> for LocalTokioExecutor
+        where
+            F: std::future::Future + 'static, // not requiring `Send`
+        {
+            fn execute(&self, fut: F) {
+                // This will spawn into the currently running `LocalSet`.
+                tokio::task::spawn_local(fut);
+            }
+        }
+
+        #[allow(unused)]
+        async fn run(io: impl crate::rt::Read + crate::rt::Write + Unpin + 'static) {
+            let (_sender, conn) =
+                crate::client::conn::http2::handshake::<_, _, http_body_util::Empty<bytes::Bytes>>(
+                    LocalTokioExecutor {
+                        _x: Default::default(),
+                    },
+                    io,
+                )
+                .await
+                .unwrap();
+
+            tokio::task::spawn_local(async move {
+                conn.await.unwrap();
+            });
+        }
+    }
+
+    #[tokio::test]
+    #[ignore] // only compilation is checked
+    async fn send_sync_executor_of_send_futures() {
+        #[derive(Clone)]
+        struct TokioExecutor;
+
+        impl<F> crate::rt::Executor<F> for TokioExecutor
+        where
+            F: std::future::Future + 'static + Send,
+            F::Output: Send + 'static,
+        {
+            fn execute(&self, fut: F) {
+                tokio::task::spawn(fut);
+            }
+        }
+
+        #[allow(unused)]
+        async fn run(io: impl crate::rt::Read + crate::rt::Write + Send + Unpin + 'static) {
+            let (_sender, conn) = crate::client::conn::http2::handshake::<
+                _,
+                _,
+                http_body_util::Empty<bytes::Bytes>,
+            >(TokioExecutor, io)
+            .await
+            .unwrap();
+
+            tokio::task::spawn(async move {
+                conn.await.unwrap();
+            });
+        }
+    }
+
+    #[tokio::test]
+    #[ignore] // only compilation is checked
+    async fn not_send_not_sync_executor_of_send_futures() {
+        #[derive(Clone)]
+        struct TokioExecutor {
+            // !Send, !Sync
+            _x: std::marker::PhantomData<std::rc::Rc<()>>,
+        }
+
+        impl<F> crate::rt::Executor<F> for TokioExecutor
+        where
+            F: std::future::Future + 'static + Send,
+            F::Output: Send + 'static,
+        {
+            fn execute(&self, fut: F) {
+                tokio::task::spawn(fut);
+            }
+        }
+
+        #[allow(unused)]
+        async fn run(io: impl crate::rt::Read + crate::rt::Write + Send + Unpin + 'static) {
+            let (_sender, conn) =
+                crate::client::conn::http2::handshake::<_, _, http_body_util::Empty<bytes::Bytes>>(
+                    TokioExecutor {
+                        _x: Default::default(),
+                    },
+                    io,
+                )
+                .await
+                .unwrap();
+
+            tokio::task::spawn_local(async move {
+                // can't use spawn here because when executor is !Send
+                conn.await.unwrap();
+            });
+        }
+    }
+
+    #[tokio::test]
+    #[ignore] // only compilation is checked
+    async fn send_not_sync_executor_of_send_futures() {
+        #[derive(Clone)]
+        struct TokioExecutor {
+            // !Sync
+            _x: std::marker::PhantomData<std::cell::Cell<()>>,
+        }
+
+        impl<F> crate::rt::Executor<F> for TokioExecutor
+        where
+            F: std::future::Future + 'static + Send,
+            F::Output: Send + 'static,
+        {
+            fn execute(&self, fut: F) {
+                tokio::task::spawn(fut);
+            }
+        }
+
+        #[allow(unused)]
+        async fn run(io: impl crate::rt::Read + crate::rt::Write + Send + Unpin + 'static) {
+            let (_sender, conn) =
+                crate::client::conn::http2::handshake::<_, _, http_body_util::Empty<bytes::Bytes>>(
+                    TokioExecutor {
+                        _x: Default::default(),
+                    },
+                    io,
+                )
+                .await
+                .unwrap();
+
+            tokio::task::spawn_local(async move {
+                // can't use spawn here because when executor is !Send
+                conn.await.unwrap();
+            });
+        }
+    }
+}
diff --git a/src/proto/h2/client.rs b/src/proto/h2/client.rs
index 6675142118..53f36a7f85 100644
--- a/src/proto/h2/client.rs
+++ b/src/proto/h2/client.rs
@@ -611,7 +611,7 @@ where
     B: Body + 'static + Unpin,
     B::Data: Send,
     B::Error: Into<Box<dyn std::error::Error + Send + Sync>>,
-    E: Http2ClientConnExec<B, T> + 'static + Send + Sync + Unpin,
+    E: Http2ClientConnExec<B, T> + Unpin,
     T: Read + Write + Unpin,
 {
     type Output = crate::Result<Dispatched>;