The characterization of $\mathbf{W_2}$ graphs by neighborhood independence

Let $G$ be a graph, let $V(G)$ denote its vertex set, let $N(v)$ and $N[v]$ denote the open and closed neighborhoods of a vertex $v$, let $N(S)$ denote the neighborhood of a vertex set $S$, and let $\alpha$ denote the independence number. The class $\mathbf{W_2}$ is the class of graphs whose independent sets can be extended to independent sets of maximum cardinality in the relevant induced subgraphs.

$\mathbf{W_2}$ characterization conjecture. For every graph $G$, the following assertions are equivalent:

1. $G$ is in the class $\mathbf{W_2}$.
2. $\alpha(N(v) \setminus N(S)) = 1$ for every vertex $v \in V(G)$ and for every maximal independent set $S$ of $V(G) \setminus N[v]$.

The equivalence is proved in the source for well-covered graphs; the conjecture asks whether the same characterization holds for arbitrary graphs.