A depth-bounded local converse conjecture for supercuspidal representations of general linear groups

Let $F$ be the underlying non-Archimedean local field, let $n$ be a positive integer, and let $\pi_{1}$ and $\pi_{2}$ be irreducible supercuspidal representations of $\operatorname{GL}(n,F)$ with the same central character. For a representation $\rho$, write $d(\rho)$ for its depth, and let $\psi_F$ be the fixed nontrivial additive character of $F$. Depth-bounded local converse conjecture. If

$$\gamma(s,\pi_{1}\times\tau,\psi_F)=\gamma(s,\pi_{2}\times\tau,\psi_F)$$

for every irreducible supercuspidal representation $\tau$ of $\operatorname{GL}(r,F)$ with $1\leq r\leq\left\lfloor\frac n2\right\rfloor$ and $d(\tau)\leq d(\pi_i)$, then $\pi_1\cong\pi_2$. This proposes that, once the central character is fixed, twists of bounded depth up to the middle rank should determine a supercuspidal representation.