Stanley's symmetry characterization conjecture for Schur labeled skew shape posets

Let $\sfP_n$ be the set of posets with underlying set $[n]=\{1,2,\ldots,n\}$. For $P\in\sfP_n$, let $K_P$ be its $P$-partition generating function, and let $\sfSP_n$ denote the class of Schur labeled skew shape posets.

Stanley's symmetry characterization conjecture. If $K_P$ is symmetric, then $P\in\sfSP_n$.

This is the implication from Stanley's broader if-and-only-if conjecture that is needed to study when modules associated with regular posets are isomorphic. The supplied text gives no resolution evidence, so the conjecture remains open here.