The Weyl-invariant generation conjecture for the Gelfand–Tsetlin geometric crystal

Let $\operatorname{GT}_m^{\leq n}$ be the Gelfand--Tsetlin geometric crystal with $Q$-pattern coordinates $z'_{j,i}$, and let the symmetric group $S_m$ act on it by the birational Weyl-group action.

Weyl-invariant generation conjecture. The subfield of $\mathbb{C}(z'_{j,i})$ fixed by this birational action is generated by the reduced $Q$-invariants and shape invariants.

This is presented as an equivalent formulation of the field-generation part of the $Q$-invariant conjecture, after transporting the geometric $R$-matrix action to the Gelfand--Tsetlin crystal through geometric RSK. It remains open in the supplied text.