The unmixed admissible clutter conjecture

Let $\mathcal{C}$ be an admissible clutter, and let $I(\mathcal{C})$ denote its edge ideal. The clutter is unmixed when all its minimal vertex covers have the same cardinality, and $I(\mathcal{C})$ is Cohen–Macaulay when its quotient ring has this property in the standard graded sense. Unmixed admissible clutter conjecture. If $\mathcal{C}$ is unmixed, then $I(\mathcal{C})$ is Cohen–Macaulay. This is suggested by the corresponding Cohen–Macaulay and unmixed criteria for bipartite graphs; the general implication for admissible clutters remains open.