The clique-cutset conjecture for minimally non-perfectly divisible graphs

Let $G$) be a graph. A graph is perfectly divisible if every induced subgraph with at least one edge has a partition $(A,B)$ such that the induced graph on $A$ is perfect and $(G[B])<(G)$. A graph is minimally non-perfectly divisible (MNPD) if it is not perfectly divisible but every proper induced subgraph is perfectly divisible. A clique cutset is a clique whose deletion disconnects $G$.

Clique-cutset conjecture. No MNPD graph contains a clique cutset.

This conjecture is proposed as an analogue of the folklore result that no minimal imperfect graph contains a clique cutset. Its status is open.