Ext-duality and generic Ext-vanishing conjecture for general linear groups

Let $F$ be a non-Archimedean local field, let $\pi_1$ and $\pi_2$ be irreducible admissible representations of ${\rm GL}_{n+1}(F)$ and ${\rm GL}_n(F)$, respectively, and let $D(\pi_2)$ denote the Aubert–Zelevinsky involution of $\pi_2$. Define $d(\pi_2)$ to be the largest integer $d$ such that ${\rm Ext}^{d}_{{\rm GL}_n(F)}[\pi_2,D(\pi_2)]\neq0$. General linear Ext-duality conjecture. The pairing

$${\rm Ext}^i_{{\rm GL}_n(F)}[\pi_2,\pi_1]\times {\rm Ext}^j_{{\rm GL}_n(F)}[\pi_1,D(\pi_2)]\longrightarrow {\rm Ext}^{i+j=d(\pi_2)}_{{\rm GL}_n(F)}[\pi_2,D(\pi_2)]\cong\mathbb C$$

is perfect. Moreover, if $\pi_1$ and $\pi_2$ are generic, then ${\rm Ext}^i_{{\rm GL}_n(F)}[\pi_1,\pi_2]=0$ for $i>0$. This specializes the preceding duality proposal to the ${\rm GL}_{n+1}\supset{\rm GL}_n$ branching setting; the paper presents it as conjectural.