Let $G$ be a graph and $M\subseteq V(G)$ be a maximal independent set. Is there a coloring $c:V(G)\to\chi(G)$ such that $c$ is constant on $M$?

(The answer is positive for graphs with infinite chromatic number: Given any coloring $c:V(G)\to\chi(G)$ define a new coloring $c'$ by $c(m) = 0$ for $m\in M$ and $c'(x) = c(x)+1$ for $x\in $V(G)\setminus M$.)