Converse conjecture for finite-field gamma factors and Gauss sums

Let $n>m$. Let $\pi$ and $\tau$ be irreducible cuspidal representations of $\operatorname{GL}_n(\mathbb F_q)$ and $\operatorname{GL}_m(\mathbb F_q)$, corresponding to regular characters $\chi$ of $\mathbb F_{q^n}^{\times}$ and $\eta$ of $\mathbb F_{q^m}^{\times}$. Let $\operatorname{Nr}_{mn:n}$ and $\operatorname{Nr}_{mn:m}$ be the norm maps, and let $G(\beta,\psi)$ denote the stated Gauss sum over $\mathbb F_{q^N}^{\times}$. The gamma-factor Gauss-sum conjecture.

$$\gamma(\pi\times\tau,\psi)=c\,\chi(-1)^{m-1}\eta(-1)^{n-1}G(\chi\circ\operatorname{Nr}_{mn:n}\cdot\eta\circ\operatorname{Nr}_{mn:m},\psi),$$

where

$$c=(-1)^{m(n-1)}q^{-mn+\frac{m^2+m}{2}}.$$

The formula is proposed as a finite-field analogue of the tensor-product gamma-factor relation and is used to study the local converse problem; the source does not state a resolution.