Polynomiality conjecture for weight multiplicities of Weyl modules

Let ${\mathfrak g}$ be a finite-dimensional simple Lie algebra with fundamental weights $\omega_i$, simple roots $\alpha_i$, rank ${\rm rk}({\mathfrak g})$, and let $P^+$ be its set of dominant integral weights. Write

$$\lambda=\sum_i\lambda_i\omega_i,\qquad \mu=\sum_i\mu^i\alpha_i.$$

For a finitely generated algebra $A$ and a character $\epsilon:A\to\mathbb C$, let $W^A_\epsilon(\lambda)$ denote the corresponding Weyl module; write $W^d(\lambda)$ for the case $A=\mathbb C[x^1,\dots,x^d]$ and evaluation at the origin, and let $M=\operatorname{Spec}A$.

Polynomiality conjecture. For $d>0$, $\dim W^d(\lambda)^{\lambda-\mu}$ is a polynomial in $\lambda_1,\dots,\lambda_{{\rm rk}({\mathfrak g})}$, of degree $\mu^i d$ in the variable $\lambda_i$ for $i=1,\dots,{\rm rk}({\mathfrak g})$. Even for a singular point, there exists $\nu$ such that, whenever $\lambda-\nu\in P^+$, $\dim W^A_\epsilon(\lambda)^{\lambda-\mu}$ is a polynomial in these variables, of degree $\mu^i d$ in $\lambda_i$ for $i=1,\dots,{\rm rk}({\mathfrak g})$, where $d=\dim M$.

This predicts that weight multiplicities of Weyl modules vary polynomially with the highest weight, including at singular points of the underlying algebraic scheme. The supplied text gives no resolution status.