diff --git a/src/Test/test_model.jl b/src/Test/test_model.jl
index 18eb8d123e..2bd53d4229 100644
--- a/src/Test/test_model.jl
+++ b/src/Test/test_model.jl
@@ -1013,3 +1013,41 @@ function test_model_delete(model::MOI.ModelLike, config::Config)
         ),
     )
 end
+
+"""
+    test_ListOfConstraintAttributesSet(
+        model::MOI.ModelLike,
+        config::Config{T},
+    ) where {T}
+
+Test issue #1429: ConstraintName should only be included in
+`ListOfConstraintAttributesSet` if it actually is set.
+"""
+function test_ListOfConstraintAttributesSet(
+    model::MOI.ModelLike,
+    ::Config{T},
+) where {T}
+    x = MOI.add_variable(model)
+    MOI.add_constraint(model, MOI.SingleVariable(x), MOI.GreaterThan(zero(T)))
+    c = MOI.add_constraint(
+        model,
+        MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(one(T), x)], zero(T)),
+        MOI.EqualTo(one(T)),
+    )
+    MOI.set(model, MOI.ConstraintName(), c, "c")
+    @test MOI.get(
+        model,
+        MOI.ListOfConstraintAttributesSet{
+            MOI.ScalarAffineFunction{T},
+            MOI.EqualTo{T},
+        }(),
+    ) == [MOI.ConstraintName()]
+    @test MOI.get(
+        model,
+        MOI.ListOfConstraintAttributesSet{
+            MOI.SingleVariable,
+            MOI.GreaterThan{T},
+        }(),
+    ) == []
+    return
+end
diff --git a/src/Utilities/model.jl b/src/Utilities/model.jl
index 30209366f5..1fad5751e1 100644
--- a/src/Utilities/model.jl
+++ b/src/Utilities/model.jl
@@ -322,9 +322,12 @@ end
 
 function MOI.get(
     model::AbstractModel,
-    ::MOI.ListOfConstraintAttributesSet,
-)::Vector{MOI.AbstractConstraintAttribute}
-    return isempty(model.con_to_name) ? [] : [MOI.ConstraintName()]
+    ::MOI.ListOfConstraintAttributesSet{F,S},
+) where {F,S}
+    if any(k -> k isa MOI.ConstraintIndex{F,S}, keys(model.con_to_name))
+        return MOI.AbstractConstraintAttribute[MOI.ConstraintName()]
+    end
+    return MOI.AbstractConstraintAttribute[]
 end
 
 # Objective
diff --git a/test/Utilities/model.jl b/test/Utilities/model.jl
index 196c9cb4d8..be2dcfdb27 100644
--- a/test/Utilities/model.jl
+++ b/test/Utilities/model.jl
@@ -593,3 +593,34 @@ end
     bound_vectors_test(Int, 0, 0)
     bound_vectors_test(Float64, -Inf, Inf)
 end
+
+@testset "ListOfConstraintAttributesSet" begin
+    model = MOI.Utilities.Model{Float64}()
+    MOI.Utilities.loadfromstring!(
+        model,
+        """
+        variables: a, b, c, d
+        minobjective: a + b + c + d
+        a >= 1.0
+        b <= 2.0
+        c == 3.0
+        d in Interval(-4.0, 4.0)
+        a in Integer()
+        b in ZeroOne()
+        c7: a + b >= -1.1
+        c8: a + b <= 2.2
+        c8: c + d == 2.2
+       """,
+    )
+    @test MOI.get(
+        model,
+        MOI.ListOfConstraintAttributesSet{MOI.SingleVariable,MOI.Integer}(),
+    ) == []
+    @test MOI.get(
+        model,
+        MOI.ListOfConstraintAttributesSet{
+            MOI.ScalarAffineFunction{Float64},
+            MOI.EqualTo{Float64},
+        }(),
+    ) == [MOI.ConstraintName()]
+end