The exact failure criterion for the naive mod $\ell$ converse theorem for $\operatorname{GL}_2(\mathbb{F}_q)$

Let $\ell$ be a prime and let $q$ be a prime power. The naive converse theorem for mod $\ell$ representations of $\operatorname{GL}_2(\mathbb{F}_q)$ asks whether the relevant mod $\ell$ gamma factors determine the representation. Naive converse theorem failure conjecture. The naive converse theorem for mod $\ell$ representations of $\operatorname{GL}_2(\mathbb{F}_q)$ fails exactly when

$$q=2\ell^i+1$$

for some value of $i>0$. The authors found counterexamples for several pairs $(\ell,q)$ satisfying this condition by explicit computation; the converse implication, and hence the exact characterization in general, remains open.