Level-rank conjecture for finite reductive groups

Keep $q$, $\mathbf{G}$, and $F$ fixed. Let $(L,\lambda)$ be a $\Phi_l$-cuspidal pair and $(M,\mu)$ a $\Phi_m$-cuspidal pair, for separate positive integers $l$ and $m$, and fix primitive roots of unity $\zeta_l$ and $\zeta_m$ of orders $l$ and $m$. Let $\operatorname{Irr}(W_{L,\lambda}^G)_{M,\mu}$ and $\operatorname{Irr}(W_{M,\mu}^G)_{L,\lambda}$ be the images of

$$\operatorname{Irr}(W_{L,\lambda}^G) \xleftarrow{\chi_{L,\lambda}} \operatorname{Uch}(G)_{L,\lambda}\cap\operatorname{Uch}(G)_{M,\mu} \xrightarrow{\chi_{M,\mu}} \operatorname{Irr}(W_{M,\mu}^G).$$

Level-rank conjecture. The two image sets are unions of $H_{L,\lambda}^G(\zeta_m)$- and $H_{M,\mu}^G(\zeta_l)$-blocks, respectively; the bijection between them induced by $\chi_{L,\lambda}$ and $\chi_{M,\mu}$ respects blocks and descends to a bijection

$$\{\text{$H_{L,\lambda}^G(\zeta_m)$-blocks $\mathbf{a}\subseteq\operatorname{Irr}(W_{L,\lambda}^G)_{M,\mu}$}\}\xrightarrow{\sim}\{\text{$H_{M,\mu}^G(\zeta_l)$-blocks $\mathbf{b}\subseteq\operatorname{Irr}(W_{M,\mu}^G)_{L,\lambda}$}\}.$$

Moreover, if $\chi_{M,\mu}^{L,\lambda}(\mathbf{a})=\mathbf{b}$, then the bijection $\mathbf{a}\xrightarrow{\sim}\mathbf{b}$ is categorified by an equivalence

$$\mathsf{D}^b(\mathsf{O}_{W_{L,\lambda}^G}(\vec{\nu}_l)_\mathbf{a})\simeq\mathsf{D}^b(\mathsf{O}_{W_{M,\mu}^G}(\vec{\nu}_m)_\mathbf{b})$$

for any vectors $\vec{\nu}_l,\vec{\nu}_m$ related to $\mathbf{a},\mathbf{b}$ by the stated numerical conditions. This proposes a level-rank correspondence simultaneously at the level of blocks and highest-weight covers; the source does not provide a resolution.