diff --git a/src/Bridges/Constraint/bridges/QuadtoSOCBridge.jl b/src/Bridges/Constraint/bridges/QuadtoSOCBridge.jl
index 23431d0645..437547c006 100644
--- a/src/Bridges/Constraint/bridges/QuadtoSOCBridge.jl
+++ b/src/Bridges/Constraint/bridges/QuadtoSOCBridge.jl
@@ -80,22 +80,32 @@ function compute_sparse_sqrt_fallback(Q, ::F, ::S) where {F,S}
 end
 
 function compute_sparse_sqrt(Q, func, set)
-    factor = try
-        LinearAlgebra.cholesky(Q; check = false)
-    catch
-        msg = "There was an error computing a Cholesky decomposition"
+    # There's a big try-catch here because Cholesky can fail even if
+    # `check = false`. As one example, it currently (v1.12) fails with
+    # `BigFloat`. Similarly, we want to guard against errors in
+    # `compute_sparse_sqrt_fallback`.
+    #
+    # The try-catch isn't a performance concern because the alternative is not
+    # being able to reformulate the problem.
+    try
+        factor = LinearAlgebra.cholesky(Q; check = false)
+        if !LinearAlgebra.issuccess(factor)
+            return compute_sparse_sqrt_fallback(Q, func, set)
+        end
+        L, p = SparseArrays.sparse(factor.L), factor.p
+        # We have Q = P' * L * L' * P. We want to find Q = U' * U, so U = L' * P
+        # First, compute L'. Note I and J are reversed
+        J, I, V = SparseArrays.findnz(L)
+        # Then, we want to permute the columns of L'. The rows stay in the same
+        # order.
+        return I, p[J], V
+    catch err
+        if err isa MOI.AddConstraintNotAllowed
+            rethrow(err)
+        end
+        msg = "There was an error computing a matrix square root"
         throw(MOI.UnsupportedConstraint{typeof(func),typeof(set)}(msg))
     end
-    if !LinearAlgebra.issuccess(factor)
-        return compute_sparse_sqrt_fallback(Q, func, set)
-    end
-    L, p = SparseArrays.sparse(factor.L), factor.p
-    # We have Q = P' * L * L' * P. We want to find Q = U' * U, so U = L' * P
-    # First, compute L'. Note I and J are reversed
-    J, I, V = SparseArrays.findnz(L)
-    # Then, we want to permute the columns of L'. The rows stay in the same
-    # order.
-    return I, p[J], V
 end
 
 function bridge_constraint(