Let $G$ be an undirected, simple graph, and let $\alpha(G)$ denote the independence number of $G$, i.e., the size of a maximum independent set (stable set) in $G$. A graph is $\alpha$-critical if for every edge $e$ of $G$, we have $\alpha(G - e) > \alpha(G)$.

Conjecture: for every $\alpha$-critical graph $G$ without isolated vertices, for every maximum independent set $S$ in $G$, there is a maximum independent set $S'$ in $G$ that is disjoint from $S$.

The conjecture is true for the simplest classes of $\alpha$-critical graphs, the graphs $K_n$ ($n \geq 2$) and the odd cycles. It also seems to be true for the list of minor-minimal obstructions to having a vertex cover of size at most five, which are also $\alpha$-critical. Is it true in general?

Thanks in advance!