Intersection-of-series block bijection conjecture

Let $\mathbb{G}$ be a generic group, let $e,m>0$, and let $(\mathbb{L},\lambda)$ and $(\mathbb{M},\mu)$ be respectively $\Phi_e$- and $\Phi_m$-cuspidal pairs. Let the intersection of their Harish–Chandra series map to subsets of the irreducible representations of the corresponding relative Weyl groups. Intersection-of-series block bijection conjecture. If the Broué–Malle–Michel specialization conjecture holds for both pairs, these subsets should be unions of the relevant blocks, and the induced bijection between the intersections should descend to a bijection between the matching blocks. This is the block-theoretic level–rank duality proposed in the paper; the supplied text gives no resolution status.