Stanley's chromatic conjecture for trees

Let $T$ and $U$ be trees, and let $X_T$ and $X_U$ denote their chromatic symmetric functions.

Stanley's chromatic conjecture for trees. Trees $T$ and $U$ are isomorphic as graphs if and only if

$$X_T=X_U.$$

This is an open problem asking whether the chromatic symmetric function distinguishes trees up to graph isomorphism.