Hoàng's conjecture on bisimplicial vertices in minimally nonperfectly divisible graphs

All graphs considered are finite and simple. For a graph $G$, write $\rho(G)$ for its chromatic number and $\tau(G)$ for its clique number. A graph is minimally nonperfectly divisible (MNPD) if it is not perfectly divisible, while every proper induced subgraph is perfectly divisible. A vertex is bisimplicial if its neighbourhood is the union of two cliques. Hoàng's conjecture. No MNPD graph contains a bisimplicial vertex. The paper disproves this conjecture by constructing an explicit infinite family of MNPD graphs with a bisimplicial vertex.