The level-scaling conjecture for two-dimensional Weyl modules

Let ${\mathfrak g}={\mathfrak gl}_r$, let $\xi$ be a partition, let $W^2(\xi)$ be the two-dimensional Weyl module, and let $V_\xi$ be the ${\mathfrak gl}_r\otimes\mathbb C\langle X,Y\rangle$-module generated by the vector defined in the source. Write ${\rm gr}_C$ for the associated graded module and $V_\xi^{[k]}$ for the level-scaled module. Two-dimensional Weyl-module conjecture. One has

$$W^2(\xi)^{[k]}\cong {\rm gr}_C\left(V_\xi^{[k]}\right).$$

The preceding theorem establishes that ${\rm gr}_C V_\xi$ is a quotient of $W^2(\xi)$, with equality when $\xi=(n)$; the conjecture asks for the corresponding equality after level scaling for every partition.