diff --git a/src/Bridges/Constraint/bridges/split_complex_equalto.jl b/src/Bridges/Constraint/bridges/split_complex_equalto.jl
index 325612a0b6..ebebe8e8cb 100644
--- a/src/Bridges/Constraint/bridges/split_complex_equalto.jl
+++ b/src/Bridges/Constraint/bridges/split_complex_equalto.jl
@@ -159,11 +159,11 @@ function MOI.delete(model::MOI.ModelLike, bridge::SplitComplexEqualToBridge)
 end
 
 function MOI.supports(
-    ::MOI.ModelLike,
-    ::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
-    ::Type{<:SplitComplexEqualToBridge},
-)
-    return true
+    model::MOI.ModelLike,
+    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    ::Type{<:SplitComplexEqualToBridge{T,F}},
+) where {T,F}
+    return MOI.supports(model, attr, MOI.ConstraintIndex{F,MOI.EqualTo{T}})
 end
 
 function MOI.get(
diff --git a/src/Bridges/Constraint/bridges/split_complex_zeros.jl b/src/Bridges/Constraint/bridges/split_complex_zeros.jl
index ba637e1faf..3e12f77136 100644
--- a/src/Bridges/Constraint/bridges/split_complex_zeros.jl
+++ b/src/Bridges/Constraint/bridges/split_complex_zeros.jl
@@ -157,11 +157,11 @@ function MOI.delete(model::MOI.ModelLike, bridge::SplitComplexZerosBridge)
 end
 
 function MOI.supports(
-    ::MOI.ModelLike,
-    ::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
-    ::Type{<:SplitComplexZerosBridge},
-)
-    return true
+    model::MOI.ModelLike,
+    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    ::Type{<:SplitComplexZerosBridge{T,F}},
+) where {T,F}
+    return MOI.supports(model, attr, MOI.ConstraintIndex{F,MOI.Zeros})
 end
 
 function MOI.get(
diff --git a/src/Bridges/Constraint/bridges/split_hyperrectangle.jl b/src/Bridges/Constraint/bridges/split_hyperrectangle.jl
index 0a573f0b4f..9128b6b043 100644
--- a/src/Bridges/Constraint/bridges/split_hyperrectangle.jl
+++ b/src/Bridges/Constraint/bridges/split_hyperrectangle.jl
@@ -154,11 +154,11 @@ function MOI.get(
 end
 
 function MOI.supports(
-    ::MOI.ModelLike,
-    ::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
-    ::Type{<:SplitHyperRectangleBridge},
-)
-    return true
+    model::MOI.ModelLike,
+    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    ::Type{<:SplitHyperRectangleBridge{T,G}},
+) where {T,G}
+    return MOI.supports(model, attr, MOI.ConstraintIndex{G,MOI.Nonnegatives})
 end
 
 function MOI.set(
diff --git a/src/Bridges/Constraint/bridges/square.jl b/src/Bridges/Constraint/bridges/square.jl
index a3bacefbdd..0d3d40d0d8 100644
--- a/src/Bridges/Constraint/bridges/square.jl
+++ b/src/Bridges/Constraint/bridges/square.jl
@@ -279,9 +279,10 @@ end
 function MOI.supports(
     model::MOI.ModelLike,
     attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
-    ::Type{<:SquareBridge},
-)
-    return true
+    ::Type{SquareBridge{T,F,G,TT,ST}},
+) where {T,F,G,TT,ST}
+    return MOI.supports(model, attr, MOI.ConstraintIndex{F,TT}) &
+           MOI.supports(model, attr, MOI.ConstraintIndex{G,MOI.EqualTo{T}})
 end
 
 function MOI.set(
diff --git a/src/Bridges/Variable/bridges/hermitian.jl b/src/Bridges/Variable/bridges/hermitian.jl
index d05da70c98..71d33e7b68 100644
--- a/src/Bridges/Variable/bridges/hermitian.jl
+++ b/src/Bridges/Variable/bridges/hermitian.jl
@@ -332,11 +332,11 @@ function unbridged_map(
 end
 
 function MOI.supports(
-    ::MOI.ModelLike,
-    ::MOI.VariablePrimalStart,
+    model::MOI.ModelLike,
+    attr::MOI.VariablePrimalStart,
     ::Type{<:HermitianToSymmetricPSDBridge},
 )
-    return true
+    return MOI.supports(model, attr, MOI.VariableIndex)
 end
 
 function MOI.set(