The bounded Möbius-action intersection conjecture

There exist constants $\varepsilon_0>0$ and $A$ such that, for every prime $p$, every multiplicative subgroup $\mathcal G\subseteq\overline{\mathbb F}_p$ with $\#\mathcal G\le p^{\varepsilon_0}$, and all elements $\alpha_{1,1},\alpha_{1,2},\alpha_{2,1},\alpha_{2,2}\in\overline{\mathbb F}_p$ satisfying

$$\alpha_{1,1}\ne0,\qquad \alpha_{1,2}\ne0,\qquad \alpha_{1,1}\alpha_{2,2}-\alpha_{1,2}\alpha_{2,1}\ne0,$$

the relevant fractional-linear equation is well-defined wherever its denominator is nonzero. Bounded Möbius-action conjecture. The equation

$$\frac{\alpha_{1,1}u-\alpha_{1,2}}{\alpha_{2,1}u-\alpha_{2,2}}=v$$

has at most $A$ solutions $(u,v)\in\mathcal G^2$. This is the conditional hypothesis used for the paper's subsequent result; the supplied text does not indicate whether it is open or resolved.