Supercuspidal form of Jacquet's local converse conjecture

Let $F$ be a locally compact non-archimedean local field, let ${\mathrm{G}}_n={\mathrm{GL}}_n(F)$, and let $\psi$ be a nontrivial additive character of $F$. Assume that $\pi_1$ and $\pi_2$ are irreducible unitarizable supercuspidal representations of ${\mathrm{G}}_n$. The supercuspidal Jacquet conjecture. If

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

as functions of the complex variable $s$ for every irreducible supercuspidal representation $\tau$ of ${\mathrm{G}}_r$ with $r=1,\ldots,\left[\frac{n}{2}\right]$, then $\pi_1$ and $\pi_2$ are equivalent as representations of ${\mathrm{G}}_n$. The source states that this formulation is equivalent to the preceding Jacquet conjecture by a standard argument, so it is a reformulation rather than an independent claim; it is included because it is separately stated as a conjecture in the paper.