Jacquet's conjecture on gamma factors for generic representations

Let $F$ be a non-archimedean local field, let $G_n={\mathrm{GL}}_n(F)$, and let $\pi_1,\pi_2$ be irreducible generic representations of $G_n$. They satisfy $\mathcal H_0$ if they have the same central character. For $m\geq1$, they satisfy $\mathcal H_m$ if they satisfy $\mathcal H_0$ and

$$\gamma(s,\pi_1\times\tau,\psi)=\gamma(s,\pi_2\times\tau,\psi)$$

as functions of $s$ for every irreducible generic representation $\tau$ of $G_m$. They satisfy $\mathcal H_{\leq r}$ if they satisfy $\mathcal H_m$ for every $0\leq m\leq r$.

Jacquet's conjecture. If $\pi_1$ and $\pi_2$ satisfy $\mathcal H_{\leq r}$, then $\pi_1\simeq\pi_2$.

This family includes the local converse conjecture as the case $r=\left[\frac n2\right]$. The source records that the cases $r=n-1$ and $r=n-2$ are proved, as are the cases $\mathcal J(3,1)$ and $\mathcal J(2,1)$; the general assertion is therefore not presented as wholly open.