Well-covered tree determination by the independence polynomial

Let $G$ be a connected graph and let $T$ be a well-covered tree. Suppose that $G$ and $T$ have the same independence polynomial:

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

Independence-polynomial conjecture. Then $G$ is a well-covered tree. This concerns whether a well-covered tree is characterized, among connected graphs, by its independence polynomial.