Ideal clique clutters of perfect graphs have the MFMC property

Let $G$ be a perfect graph, and let $\mathcal{C}$ be its clique clutter, whose edges are the maximal cliques of $G$. The clutter $\mathcal{C}$ is ideal when its covering polyhedron has only integral vertices, and it has the max-flow min-cut (MFMC) property in the usual clutter sense. Perfect-graph clique-clutter conjecture. If $\mathcal{C}$ is ideal, then $\mathcal{C}$ has the MFMC property. The conjecture is a proposed weakening of the Conforti–Cornuéjols packing-property conjecture, motivated by the fact that idealness is necessary for MFMC and by known sufficient conditions such as the diadic property; the source gives no resolution.