Strong and powerful tableau bounds for chromatic symmetric function coefficients

Let $P$) be a $(\mathbf{3}+\mathbf{1})$-free poset, let $\lambda$ be a partition, and let $c_\lambda^P$ denote the coefficient of $e_\lambda(\mathbf{x})$ in $X_{\operatorname{inc}(P)}(\mathbf{x})$. Let $\operatorname{strongST}_P(\lambda)$ and $\operatorname{powST}_P(\lambda)$ be the sets of strong and powerful standard $P$-tableaux of shape $\lambda$. Strong and powerful tableau bounds conjecture. For a $(\mathbf{3}+\mathbf{1})$-free poset $P$,

$$\#\operatorname{strongST}_P(\lambda)\leq c_\lambda^P\leq\#\operatorname{powST}_P(\lambda).$$

These conjectured bounds seek a combinatorial interpretation of the elementary-basis coefficients of chromatic symmetric functions. The paper also proposes refinements for natural unit interval orders and reports verification in small cases.