The strict P-partition enumerator conjecture for tree posets

Let $P$ and $Q$ be finite posets whose Hasse diagrams are trees, and let $\overline{K}_{P}(\mathbf{x})$ denote the strict $P$-partition enumerator. The strict P-partition enumerator conjecture. If $P$ and $Q$ are not isomorphic, then

$$\overline{K}_{P}(\mathbf{x})\neq \overline{K}_{Q}(\mathbf{x}).$$

The conjecture would imply the chromatic quasisymmetric function conjecture for directed trees, because $\overline{K}_{P}(\mathbf{x})$ is the coefficient of the highest power of $t$ in the chromatic quasisymmetric function associated with a directed acyclic graph. It has been verified for all posets with at most 11 elements and appears as a question in work of Hamaker and Tseng; the general case remains open.