diff --git a/test/Bridges/Variable/kernel.jl b/test/Bridges/Variable/kernel.jl
index b6ed4def5..e82aa67b9 100644
--- a/test/Bridges/Variable/kernel.jl
+++ b/test/Bridges/Variable/kernel.jl
@@ -22,29 +22,48 @@ function test_runtests()
     MOI.Bridges.runtests(
         SumOfSquares.Bridges.Variable.KernelBridge,
         model -> begin
-            MOI.add_constrained_variables(
+            p, _ = MOI.add_constrained_variables(
                 model,
                 SumOfSquares.WeightedSOSCone{
                     MOI.PositiveSemidefiniteConeTriangle,
                 }(
                     MB.SubBasis{MB.Monomial}([
-                        x^4,
-                        x^3 * y,
-                        x^2 * y^2,
-                        x * y^3,
                         y^4,
+                        x * y^3,
+                        x^2 * y^2,
+                        x^3 * y,
+                        x^4,
                     ]),
-                    [MB.SubBasis{MB.Monomial}([x^2, y^2, x * y])],
+                    [MB.SubBasis{MB.Monomial}([y^2, x * y, x^2])],
                     [MB.algebra_element(1.0 * x^0 * y^0)],
                 ),
             )
+            a = float.(1:length(p))
+            MOI.add_constraint(
+                model,
+                MOI.Utilities.vectorize([a' * p]),
+                MOI.Zeros(1),
+            )
         end,
         model -> begin
-            Q, _ = MOI.add_constrained_variables(
+            q, _ = MOI.add_constrained_variables(
                 model,
                 MOI.PositiveSemidefiniteConeTriangle(3),
             )
-        end,
+            a = float.(1:length(q))
+            MOI.add_constraint(
+                model,
+                MOI.Utilities.vectorize([
+                    1.0 * q[1] +
+                    4.0 * q[2] +
+                    3.0 * (1.0q[3] + 2.0q[4]) +
+                    8.0 * q[5] +
+                    5.0 * q[6],
+                ]),
+                MOI.Zeros(1),
+            )
+        end;
+        cannot_unbridge = true,
     )
     return
 end