Primitive-representation local converse conjecture over finite fields

Let $r$ be a prime divisor of $n$. For a regular character $\chi$ of $\mathbb F_{q^n}^{\times}$, let $\pi_\chi$ be the associated primitive cuspidal representation of $\operatorname{GL}_r(\mathbb F_{q^{n/r}})$. Primitive-representation local converse conjecture. If regular characters $\chi_1,\chi_2$ satisfy

$$\gamma(\pi_{\chi_1}\times\eta,\psi)=\gamma(\pi_{\chi_2}\times\eta,\psi)$$

for every character $\eta$ of $\mathbb F_{q^{n/r}}^{\times}$, then $\pi_{\chi_1}\cong\pi_{\chi_2}$. This proposes a restricted local converse theorem using twists by characters; the source gives no proof or resolution.