Levit–Mandrescu conjecture on independence-equivalent well-covered trees

Let $G$ be a connected graph, let $T$ be a well-covered tree, and let $I(G,x)$ denote the independence polynomial of $G$. Levit–Mandrescu conjecture. If

$$I(T,x)=I(G,x),$$

then $G$ is a well-covered tree. The conjecture concerns whether independence-polynomial equivalence with a well-covered tree forces the other graph to share that structure; the paper later gives an infinite family of counterexamples, so the conjecture is refuted.