Equality-case conjecture for cross-1-intersecting subsets of $\mathrm{GL}_2(q)$

Fix an odd prime power $q$ and factor

$$q-1=\ell m,$$

where $\ell$ is an odd prime with $\ell\nmid m$. Let $\beta:\mathrm{GL}_2(q)\to\mathrm{GL}(V_{m,0})$ be the irreducible principal series representation, and let $\log_\varepsilon:\mathbb{F}_q^\times\to\mathbb{Z}/(q-1)\mathbb{Z}$ be the discrete logarithm for a fixed generator $\varepsilon$ of $\mathbb{F}_q^\times$. Two subsets $S_1,S_2\subseteq\mathrm{GL}_2(q)$ are cross-1-intersecting with respect to $\beta$ if they satisfy the intersection condition defined by the representation $\beta$. Define

$$H=\left\{g\in\mathrm{GL}_2(q)\mid \log_\varepsilon(\det(g))\equiv0\pmod{\ell}\right\}.$$

Equality-case conjecture. If $S_1$ and $S_2$ are cross-1-intersecting with respect to $\beta$ and

$$\sqrt{|S_1||S_2|}=\frac{|\mathrm{GL}_2(q)|}{\ell},$$

then $S_1=S_2=H$. The theorem preceding this conjecture establishes the corresponding extremal bound, while the conjecture asserts that equality has this unique form.