diff --git a/pyomo/contrib/solver/solvers/highs.py b/pyomo/contrib/solver/solvers/highs.py
index fb83117bb81..d1bcbae8ab6 100644
--- a/pyomo/contrib/solver/solvers/highs.py
+++ b/pyomo/contrib/solver/solvers/highs.py
@@ -750,7 +750,30 @@ def _postsolve(self, stream: io.StringIO):
                     results.objective_bound = None
             else:
                 results.objective_bound = info.mip_dual_bound
-            results.iteration_count = info.simplex_iteration_count
+
+            if info.valid:
+                # The method that ran will have a non-negative iteration count
+                # and the others will be 0 or -1.
+                counts = [
+                    info.simplex_iteration_count,
+                    info.ipm_iteration_count,
+                    info.mip_node_count,
+                    info.pdlp_iteration_count,
+                    info.qp_iteration_count,
+                ]
+                positive_iters = [c for c in counts if c > 0]
+                if not positive_iters:
+                    assert any(
+                        (c == 0 for c in counts)
+                    ), "At least one iteration count should have a non-negative value"
+                    results.iteration_count = 0
+                else:
+                    assert (
+                        len(positive_iters) == 1
+                    ), "Only one iteration count should have a positive value"
+                    results.iteration_count = positive_iters[0]
+            else:
+                results.iteration_count = 0
 
         if config.load_solutions:
             if has_feasible_solution:
diff --git a/pyomo/contrib/solver/tests/solvers/test_highs.py b/pyomo/contrib/solver/tests/solvers/test_highs.py
index f59a0bfa42d..7deb4a3f310 100644
--- a/pyomo/contrib/solver/tests/solvers/test_highs.py
+++ b/pyomo/contrib/solver/tests/solvers/test_highs.py
@@ -13,6 +13,7 @@
 import pyomo.environ as pyo
 
 from pyomo.contrib.solver.solvers.highs import Highs
+from pyomo.contrib.solver.common.results import SolutionStatus
 
 opt = Highs()
 if not opt.available():
@@ -109,3 +110,62 @@ def test_fix_and_unfix(self):
         self.assertAlmostEqual(m.fx.value, 1, places=5)
         self.assertAlmostEqual(m.fy.value, 0, places=5)
         self.assertAlmostEqual(r.objective_bound, 0.5, places=5)
+
+
+class TestHighsMiniDemos(unittest.TestCase):
+    def test_lp_methods(self):
+        for method in ("simplex", "ipm", "pdlp"):
+            # Build LP
+            m = pyo.ConcreteModel()
+            m.x = pyo.Var(domain=pyo.NonNegativeReals)
+            m.y = pyo.Var(domain=pyo.NonNegativeReals)
+            m.c = pyo.Constraint(expr=m.x + m.y >= 1)
+            m.obj = pyo.Objective(expr=m.x + m.y, sense=pyo.minimize)
+
+            solver = Highs()
+            solver.config.solver_options["solver"] = (
+                method  # 'simplex' | 'ipm' | 'pdlp'
+            )
+            solver.config.solver_options["presolve"] = "off"
+            results = solver.solve(m)
+            self.assertTrue(results.solution_status == SolutionStatus.optimal)
+            self.assertTrue(results.iteration_count > 0)
+
+    def test_mip_demo(self):
+        # Build MIP
+        m = pyo.ConcreteModel()
+        m.a = pyo.Var(domain=pyo.Binary)
+        m.b = pyo.Var(domain=pyo.Binary)
+        m.cap = pyo.Constraint(expr=m.a + m.b <= 1)
+        m.obj = pyo.Objective(expr=3 * m.a + 2 * m.b, sense=pyo.maximize)
+
+        solver = Highs()
+        solver.config.solver_options["presolve"] = "off"
+        results = solver.solve(m)
+        self.assertTrue(results.solution_status == SolutionStatus.optimal)
+        self.assertTrue(results.iteration_count > 0)
+
+    def test_mip_pmedian(self):
+        # Build MIP
+        from pyomo.core.tests.examples.pmedian_concrete import create_model
+
+        M = create_model()
+
+        solver = Highs()
+        results = solver.solve(M)
+        self.assertEqual(results.solution_status, SolutionStatus.optimal)
+        self.assertTrue(results.iteration_count > 0)
+
+    def test_qp_demo(self):
+        # Build convex QP
+        m = pyo.ConcreteModel()
+        m.x = pyo.Var(domain=pyo.Reals, bounds=(0, None))
+        m.y = pyo.Var(domain=pyo.Reals, bounds=(0, None))
+        m.c = pyo.Constraint(expr=m.x + m.y >= 1)
+        m.obj = pyo.Objective(expr=(m.x - 1) ** 2 + (m.y - 2) ** 2, sense=pyo.minimize)
+
+        solver = Highs()
+        solver.config.solver_options["presolve"] = "off"
+        results = solver.solve(m)
+        self.assertTrue(results.solution_status == SolutionStatus.optimal)
+        self.assertTrue(results.iteration_count > 0)