Strong-tableau nonvanishing conjecture for elementary coefficients

Let $P$ be a $(\mathbf{3}+\mathbf{1})$-free poset, let $\lambda$ be a partition, and let $c_\lambda^P$ be the corresponding elementary-basis coefficient of $X_{\operatorname{inc}(P)}(\mathbf{x})$. Let $\operatorname{strongST}_P(\lambda)$ denote the set of strong standard $P$-tableaux of shape $\lambda$. Strong-tableau nonvanishing conjecture. If $\operatorname{strongST}_P(\lambda)$ is empty, then $c_\lambda^P=0$. The source explains that this is unresolved in general and, for natural unit interval orders, is equivalent to a formulation involving Hikita tableaux.