Zuber's polynomiality conjecture for FPL configurations with nested arches

Let $X$ be a fixed non-crossing matching with $n-m$ arches, and let $X\cup m$ be the matching obtained from $X$ by adding $m$ nested arches. Let $A_X(m)$ denote the number of fully packed loop configurations having $X\cup m$ as their associated matching. Associate to $X$ a Ferrers diagram $\lambda(X)$, and write $|\lambda(X)|$ for its size and $\dim(\lambda(X))$ for the dimension of the irreducible representation of the symmetric group $S_{|\lambda(X)|}$ indexed by it.

Zuber's conjecture. The number $A_X(m)$ is

$$A_X(m)=\frac{1}{|X|!}P_X(m),$$

where $P_X(m)$ is a polynomial of degree $|\lambda(X)|$ with integer coefficients, whose highest-degree coefficient is $\dim(\lambda(X))$.

This conjecture gives a precise form of the expected polynomial dependence on the number of added nested arches; the supplied text does not establish its resolution.