Prasad's Ext-vanishing conjecture for generic representations of general linear groups

Let $F$ be the underlying nonarchimedean local field, let $n$ be a positive integer, let $\pi_1$ be an irreducible generic representation of $GL(n+1,F)$, and let $\pi_2$ be an irreducible generic representation of $GL(n,F)$. The groups $\operatorname{Ext}^i_{GL(n,F)}(\pi_1,\pi_2)$ are taken in the category of smooth representations of $GL(n,F)$. Prasad's conjecture.

$$\operatorname{Ext}^i_{GL(n,F)}(\pi_1,\pi_2)=0$$

for all $i\geq 1$. This conjecture concerns higher extension groups in the restriction from $GL(n+1,F)$ to $GL(n,F)$; the paper verifies it for representations that are locally nice at the relevant maximal ideal, while the general case is not resolved here.