Zuber's two-matching polynomiality conjecture for FPL configurations

Let $X$ and $Y$ be two non-crossing matchings. Place them on the circle together with $m$ nested arches between them, and denote the resulting matching by $X\cup m\cup Y$. Let $\lambda(X)$ and $\lambda(Y)$ be the associated Ferrers diagrams, with sizes $|\lambda(X)|$ and $|\lambda(Y)|$, and let $\dim(\lambda(X))$ and $\dim(\lambda(Y))$ denote the dimensions of the corresponding irreducible representations of the symmetric groups indexed by these diagrams. Let $A_{X,Y}(m)$ be the number of fully packed loop configurations having $X\cup m\cup Y$ as their associated matching.

Zuber's conjecture. The number $A_{X,Y}(m)$ is

$$A_{X,Y}(m)=\frac{1}{|\lambda(X)|!\,|\lambda(Y)|!}P_{X,Y}(m),$$

where $P_{X,Y}(m)$ is a polynomial of degree $|\lambda(X)|+|\lambda(Y)|$ with integer coefficients, whose highest-degree coefficient is $\dim(\lambda(X))\cdot\dim(\lambda(Y))$.

This conjecture generalizes Zuber's first conjecture to nested arches placed between two matchings; the supplied text does not establish its resolution.