diff --git a/ext/MathOptInterfaceLDLFactorizationsExt.jl b/ext/MathOptInterfaceLDLFactorizationsExt.jl
index 9211069776..c3b5c7a31c 100644
--- a/ext/MathOptInterfaceLDLFactorizationsExt.jl
+++ b/ext/MathOptInterfaceLDLFactorizationsExt.jl
@@ -21,7 +21,11 @@ function MOI.Bridges.Constraint.compute_sparse_sqrt_fallback(
 ) where {F<:MOI.ScalarQuadraticFunction,S<:MOI.AbstractSet}
     n = LinearAlgebra.checksquare(Q)
     factor = LDLFactorizations.ldl(Q)
-    if minimum(factor.D) < 0
+    # Ideally we should use `LDLFactorizations.factorized(factor)` here, but it
+    # has some false negatives. Instead we check that the factorization appeared
+    # to work. This is a heuristic. There might be other cases where check is
+    # insufficient.
+    if minimum(factor.D) < 0 || any(issubnormal, factor.D)
         msg = """
         Unable to transform a quadratic constraint into a SecondOrderCone
         constraint because the quadratic constraint is not convex.
diff --git a/test/Bridges/Constraint/test_QuadtoSOCBridge.jl b/test/Bridges/Constraint/test_QuadtoSOCBridge.jl
index 00b64a3ff1..d34cfe5093 100644
--- a/test/Bridges/Constraint/test_QuadtoSOCBridge.jl
+++ b/test/Bridges/Constraint/test_QuadtoSOCBridge.jl
@@ -363,11 +363,11 @@ function test_compute_sparse_sqrt_edge_cases()
         [1.0 0.0; 0.0 2.0],
         # Cholesky works, with pivoting
         [1.0 0.0 1.0; 0.0 1.0 1.0; 1.0 1.0 3.0],
-        # Cholesky fails due to 0 eigen value
+        # Cholesky fails due to 0 eigen value. LDL works
         [1.0 1.0; 1.0 1.0],
         # Cholesky succeeds, even though 0 eigen value
         [2.0 2.0; 2.0 2.0],
-        # Cholesky fails because of 0 column/row
+        # Cholesky fails because of 0 column/row. LDL works
         [2.0 0.0; 0.0 0.0],
     ]
         B = SparseArrays.sparse(A)
@@ -378,12 +378,18 @@ function test_compute_sparse_sqrt_edge_cases()
         end
         @test isapprox(A, U' * U; atol = 1e-10)
     end
-    A = [-1.0 0.0; 0.0 1.0]
-    B = SparseArrays.sparse(A)
-    @test_throws(
-        MOI.UnsupportedConstraint{typeof(f),typeof(s)},
-        MOI.Bridges.Constraint.compute_sparse_sqrt(B, f, s),
-    )
+    # Test failures
+    for A in [
+        [-1.0 0.0; 0.0 1.0],
+        # Found from test_quadratic_nonconvex_constraint_basic
+        [0.0 -1.0; -1.0 0.0],
+    ]
+        B = SparseArrays.sparse(A)
+        @test_throws(
+            MOI.UnsupportedConstraint{typeof(f),typeof(s)},
+            MOI.Bridges.Constraint.compute_sparse_sqrt(B, f, s),
+        )
+    end
     return
 end