Polynomiality conjecture for lexicographic Gelfand–Tsetlin pattern counts

Fix partitions ${\lambda}/{\mu}$ and ${\nu}$, let $G$ be an element of $GT_{{\lambda}/{\mu},{\nu}}$, and define

$$p_G(k)=\left|\left\{G'\in GT_{k{\lambda}/k{\mu},k{\nu}}\mid G'\leq_{\operatorname*{lex}} kG\right\}\right|.$$

Here $GT_{{\lambda}/{\mu},{\nu}}$ denotes the set of Gelfand–Tsetlin patterns associated with the skew shape and weight, and $\leq_{\operatorname*{lex}}$ is lexicographic order. Polynomiality conjecture. For any fixed $G\in GT_{{\lambda}/{\mu},{\nu}}$, the function $p_G(k)$ is polynomial in $k$. Ehrhart theory gives quasi-polynomiality because these functions count lattice points in rational polytopes, while polynomiality is the open strengthening motivating the conjecture.