Levit–Tankus characterization conjecture for \mathbf{W_2} graphs

Let $G$ be a graph. For a vertex $v$, let $N(v)$ denote its open neighborhood and let $N[v]$ denote its closed neighborhood. Write $G-N[v]$ for the graph obtained by deleting $v$ and its neighbors, and for a set $S$ write $N(S)$ for its neighborhood. A set of vertices is independent if no two of its vertices are adjacent, and it is maximal independent if it is not properly contained in a larger independent set.

Levit–Tankus conjecture. The graph $G$ is $\mathbf{W_2}$ if and only if, for every vertex $v$ in $G$ and every maximal independent set $S$ in $G-N[v]$, the largest independent set in $N(v)\setminus N(S)$ consists of a single vertex.

The paper states this as a proposed characterization of $\mathbf{W_2}$ graphs and refutes it in the cited work, so the claim is not valid in general.