diff --git a/.github/julia/runtests_uno_filtersqp_highs.jl b/.github/julia/runtests_uno_filtersqp_highs.jl
index afaca3308..97564fc52 100644
--- a/.github/julia/runtests_uno_filtersqp_highs.jl
+++ b/.github/julia/runtests_uno_filtersqp_highs.jl
@@ -23,8 +23,8 @@ function Optimizer(options)
 end
 
 # Note: the HiGHS QP solver tolerates only positive semi-definite Hessians.
-# We therefore run only on convex instances (for now)
-Optimizer_Uno_filtersqp() = Optimizer(["logger=SILENT", "preset=filtersqp", "QP_solver=HiGHS", "max_iterations=10000", "unbounded_objective_threshold=-1e15"])
+# We therefore use primal regularization.
+Optimizer_Uno_filtersqp() = Optimizer(["logger=SILENT", "preset=filtersqp", "regularization_strategy=primal", "QP_solver=HiGHS", "max_iterations=10000", "unbounded_objective_threshold=-1e15"])
 
 # This testset runs https://github.com/jump-dev/MINLPTests.jl
 @testset "MINLPTests" begin
@@ -34,15 +34,103 @@ Optimizer_Uno_filtersqp() = Optimizer(["logger=SILENT", "preset=filtersqp", "QP_
     )
     primal_tol = 1e-4
     objective_tol = 1e-4
+    # This function tests (potentially) non-convex nonlinear programs. The tests
+    # are meant to be "easy" in the sense that most NLP solvers can find the
+    # same global minimum, but a test failure can sometimes be allowed.
+    MINLPTests.test_nlp_expr(
+        Optimizer_Uno_filtersqp;
+        exclude = [
+            "003_014",  # Local solution
+            "004_010",  # Local solution
+            "004_011",  # Local solution
+            # Remove once https://github.com/cvanaret/Uno/issues/39 is fixed
+            "005_010",
+            "007_010",
+            "008_010",  # Local solution
+            # Okay to exclude forever: AmplNLWriter does not support
+            # user-defined functions.
+            "006_010",
+        ],
+        primal_target,
+    )
     # This function tests convex nonlinear programs. Test failures here should
     # never be allowed, because even local NLP solvers should find the global
     # optimum.
     MINLPTests.test_nlp_cvx_expr(Optimizer_Uno_filtersqp;
         exclude = [
-            "102_010", "105_010", "201_011", "501_010", # negative curvature
             "110_010" # HiGHS cannot solve the first QP (https://github.com/ERGO-Code/HiGHS/issues/2489)
         ],
         primal_target,
         primal_tol,
         objective_tol)
 end
+
+# This testset runs the full gamut of MOI.Test.runtests. There are a number of
+# tests in here with weird edge cases, so a variety of exclusions are expected.
+@testset "MathOptInterface.test" begin
+    optimizer = MOI.instantiate(
+        Optimizer_Uno_filtersqp;
+        with_cache_type = Float64,
+        with_bridge_type = Float64,
+    )
+    MOI.Test.runtests(
+        optimizer,
+        MOI.Test.Config(
+            # These are pretty loose tolerances so that all tests pass. There
+            # are few tests with weird numerics. If tests fail because of
+            # tolerances, it might be okay to make these looser, or you could
+            # tighten the tolerances used by Uno.
+            atol = 1e-4,
+            rtol = 1e-4,
+            optimal_status = MOI.LOCALLY_SOLVED,
+            infeasible_status = MOI.LOCALLY_INFEASIBLE,
+            exclude = Any[
+                # It's okay to exclude BasisStatus, since AmplNLWriter doesn't
+                # support this information, so too with ObjectiveBound, since
+                # Uno is a local NLP solver.
+                MOI.VariableBasisStatus,
+                MOI.ConstraintBasisStatus,
+                MOI.ObjectiveBound,
+            ],
+        );
+        exclude = [
+            # ==================================================================
+            # The following tests are bugs.
+            #
+            # We should fix issues in Uno, and then try removing these lines.
+            #
+            # These tests return OTHER_LIMIT instead of LOCALLY_INFEASIBLE. It
+            # might be acceptable to leave this as-is, but it would be better to
+            # fix.
+            r"^test_conic_NormInfinityCone_INFEASIBLE$",
+            r"^test_conic_NormOneCone_INFEASIBLE$",
+            r"^test_conic_linear_INFEASIBLE$",
+            r"^test_conic_linear_INFEASIBLE_2$",
+            r"^test_linear_INFEASIBLE$",
+            r"^test_linear_INFEASIBLE_2$",
+            r"^test_quadratic_SecondOrderCone_basic$",
+            r"^test_quadratic_nonconvex_constraint_basic$",
+            r"^test_linear_DUAL_INFEASIBLE$",
+            r"^test_linear_DUAL_INFEASIBLE_2$",
+            r"^test_solve_TerminationStatus_DUAL_INFEASIBLE$",
+            r"^test_solve_DualStatus_INFEASIBILITY_CERTIFICATE_",
+            # ==================================================================
+            # The following tests are okay to exclude forever.
+            #
+            # Uno does not support integrality, and AmplNLWriter has no way of
+            # telling if a particular binary supports integers or not (it
+            # defaults to assuming yes).
+            "Indicator",
+            r"[Ii]nteger",
+            "Semicontinuous",
+            "Semiinteger",
+            "SOS1",
+            "SOS2",
+            "ZeroOne",
+            r"^test_cpsat_",
+            r"^test_attribute_SolverVersion$",
+            r"^test_nonlinear_invalid$",
+            r"^test_basic_VectorNonlinearFunction_",
+        ],
+    )
+end