The conjectural equivalence of simple Bernstein blocks for related general linear groups

Retain the assumptions on $F$ and $R$. Fix $k,m\geq 1$, and set

$$G_1=\operatorname{GL}_{2km}(F),\qquad G_2=\operatorname{GL}_{2k}(F^m),$$

where $F^m$ is the unramified extension of $F$ of degree $m$. Let $P_i=M_iU_i$ be standard parabolic subgroups with $M_i\cong H_i\times H_i$, where $H_1=\operatorname{GL}_{km}(F)$ and $H_2=\operatorname{GL}_k(F^m)$. For irreducible supercuspidal level-$0$ representations $\pi_i\in\operatorname{Rep}_R(H_i)$, write $\pi_i^2=\pi_i\boxtimes\pi_i\in\operatorname{Rep}_R(M_i)$.

Conjectural equivalence of Bernstein blocks. There is an equivalence of categories

$$\mathfrak{R}^{[M_1,\pi_1^2]_{G_1}}_R(G_1)\cong\mathfrak{R}^{[M_2,\pi_2^2]_{G_2}}_R(G_2).$$

This proposes that the indicated simple Bernstein blocks for the two related general linear groups are equivalent. The source notes that the case $k=1$ is the most interesting and that the general case can be obtained by applying the conjecture twice in the $k=1$ version; no resolution is supplied here.