The finite-field exceptional-set estimate in the plane

Let $A\subset \mathbb{F}_p^2$ satisfy

$$\#A=p^a.$$

For $0<s<\max\{1,a\}$, let $E_s(A)$ be the exceptional set defined for projection to lines, with $n=1$ and $k=1$. The finite-field exceptional-set conjecture. Then

$$\#E_s(A)\leq C_{\epsilon,a,s}p^{\epsilon+\max\{0,2s-a\}}.$$

This is the finite-field counterpart of the planar Euclidean projection conjecture and allows an $\epsilon$-loss in the exponent. The source does not provide evidence of resolution, so the conjecture remains open.