Mkrtchyan–Petrosyan–Vardanyan conjecture on maximum matchings

Let $G$ be a graph, possibly with multiple edges but no loops, and let $\Delta(G)$ and $\delta(G)$ denote its maximum and minimum degrees. A matching is maximum if it has largest possible cardinality, and a vertex is $M$-unsaturated if it is not incident with an edge of $M$.

Mkrtchyan–Petrosyan–Vardanyan conjecture. If

$$\Delta(G)-\delta(G)\le 1,$$

then $G$ contains a maximum matching $M$ such that no two $M$-unsaturated vertices have a common neighbor.

The conjecture is refuted: Picouleau found a counterexample that is a simple bipartite graph with $\delta(G)=4$ and $\Delta(G)=5$.