  In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with
$2\leq \delta(G)\leq \Delta(G)\leq 3$ contains a maximum matching whose
unsaturated vertices do not have a common neighbor, where $\Delta(G)$ and
$\delta(G)$ denote the maximum and minimum degrees of vertices in $G$,
respectively. In the same paper they suggested the following conjecture: every
graph $G$ with $\Delta(G)-\delta(G)\leq 1$ contains a maximum matching whose
unsaturated vertices do not have a common neighbor. Recently, Picouleau
disproved this conjecture by constructing a bipartite counterexample $G$ with
$\Delta(G)=5$ and $\delta(G)=4$. In this note we show that the conjecture is
false for graphs $G$ with $\Delta(G)-\delta(G)=1$ and $\Delta(G)\geq 4$, and
for $r$-regular graphs when $r\geq 7$.
