Polynomiality conjecture for fibres of the Gelfand–Tsetlin pattern map

Fix partitions ${\lambda}/{\mu}$ and ${\nu}$. Let $\psi_k:GT_{k{\lambda}/k{\mu},k{\nu}}\to GT_{{\lambda}/{\mu},{\nu}}$ be the map sending a pattern to the unique lexicographically largest pattern $G'$ satisfying $kG'\leq_{\operatorname*{lex}}G$. For a fixed $G\in GT_{{\lambda}/{\mu},{\nu}}$, consider the fibre $\psi_k^{-1}(G)$. Fibre polynomiality conjecture. For any fixed $G\in GT_{{\lambda}/{\mu},{\nu}}$, the size of the inverse image of $G$ under $\psi_k^{-1}$ is a polynomial in $k$. This is a later formulation involving fibres of the map $\psi_k$; the source does not provide a proof or resolution.