Strong tableau lower bound for q-chromatic coefficients

Let $P$ be a natural unit interval order with $n$ elements, let $\lambda\vdash n$, and let $\operatorname{strongST}_P(\lambda)$ be the set of strong standard $P$-tableaux of shape $\lambda$. Let $\operatorname{inv}_P(T)$ denote the tableau inversion statistic. Strong tableau q-undercount conjecture. The polynomial

$$c_\lambda^P(q)-\sum_{T\in\operatorname{strongST}_P(\lambda)}q^{\operatorname{inv}_P(T)}$$

has non-negative integer coefficients. This conjecture provides a proposed combinatorial lower bound for the $q$-elementary coefficient; the source reports verification for natural unit interval orders with at most $10$ elements.