The independence-polynomial characterization of well-covered trees

Let $T$ be a well-covered tree and let $G$ be a graph. Suppose their independence polynomials agree:

$$I(T;x)=I(G;x).$$

A graph is well-covered if all its maximal stable sets have the same cardinality.

Independence-polynomial characterization conjecture. If $T$ is a well-covered tree and $I(T;x)=I(G;x)$, then $G$ is well-covered.

The paper motivates this conjecture by examples of claw-free graphs that are both well-covered and have the same independence polynomials as the well-covered trees under investigation. The source provides no resolution status.