Degenerate supergroup conjecture for the geometric Langlands limit

Let $N>M$, let $\Gr_{\GL_N}$ be the affine Grassmannian, and let $\ul\GL(M|N)$ be the group introduced in the source as the appropriate extension governing the $c\to\infty$ limit. Let $U_{M,N}(\bF)$ and $\chi_{M,N}$ be as above. Degenerate supergroup conjecture. For $N>M$,

$$D^{(\GL(M,\bO)\ltimes U_{M,N}(\bF),\chi_{M,N})}(\Gr_{\GL_N})$$

is equivalent to the category of modules over the group $\ul\GL(M|N)$. For $N=M$,

$$D^{\GL(N,\bO)}(\Gr_{\GL_N}\times\bV)$$

is equivalent to the category of modules over $\ul\GL(N|N)$. These equivalences should hold for both derived and abelian categories. This conjecture describes the proposed $c\to\infty$ limit, where twisted $D$-modules become ordinary $D$-modules; the source explains that the ordinary supergroup $\GL(M|N)$ is not the correct limiting object and instead uses the degenerate group $\ul\GL(M|N)$.