The finite-field planar exceptional-set conjecture

Let $p$ be prime, let $X\subset\mathbb F_p^2$, and let $E_s(X)$ denote the exceptional set of projection directions with projected dimension at most $s$. Finite-field exceptional-set conjecture. For every $\epsilon>0$,

$$|E_s(X)|\lesssim p^\epsilon\max\{s^2|X|^{-1},1\}.$$

The conjecture is presented as a finite-field analogue of a continuum exceptional-set estimate. The source explicitly notes that the corresponding statement can fail over non-prime finite fields, while the status over $\mathbb F_p$ is not otherwise resolved in the supplied text.