Stanley's conjecture on symmetric P-partition enumerators

A labelled poset $P$ has a $P$-partition enumerator $F_P({\mathbf{x}})$, which is always quasisymmetric; a labelling is column-strict when it has the column-strict property used for labelled posets, and a skew Ferrers poset is the poset associated with a skew Ferrers diagram $\lambda/\mu$. Stanley's conjecture. The function $F_P({\mathbf{x}})$ is symmetric, and not merely quasisymmetric, if and only if $P$ is a column-strict labelling of some skew Ferrers poset $\lambda/\mu$. This conjecture asks for a characterization of the labelled posets whose $P$-partition enumerators are symmetric; it is attributed to Stanley's thesis and is presented in the source as an open question.