The principal specialization conjecture for tree posets

Let $P$ and $Q$ be posets with $n$ elements whose Hasse diagrams are trees. Let $\overline{K}_{P}(\mathbf{x})$ denote the strict $P$-partition enumerator, and specialize $x_i=q^{i-1}$ for $1\leq i\leq n$ and $x_i=0$ thereafter. The principal specialization conjecture for tree posets. If $P$ and $Q$ are not isomorphic, then

$$\overline{K}_{P}(1,q,q^2,\ldots,q^{n-1})\neq \overline{K}_{Q}(1,q,q^2,\ldots,q^{n-1}).$$

The conjecture has been verified for all posets with at most 10 elements. The source further reports that the shorter specialization through $q^{n-2}$ appears sufficient, but presents that stronger-looking assertion only as an observation rather than a separate conjecture.