Uniform-distribution conjecture for the lifted curve points

For each prime $p$, let $\mathcal{F}_p\subseteq\mathbb{F}_p^3$ be the set of lifted points associated with the points of $C_p$, and let $m=|\mathcal{F}_p|$. Define the discrepancy

$$\mathcal{D}_p(\mathcal{F}_p)=\sup_{E\subseteq\mathbb{F}_p^3}\left|\frac{|\mathcal{F}_p\cap E|}{|E|}\frac{p^3}{m}-1\right|.$$

Uniform-distribution conjecture. As $p\to\infty$,

$$\mathcal{D}_p(\mathcal{F}_p)\to 0.$$

This conjecture is presented as a uniform-distribution condition implying the rank-two tuple asymptotic above. The supplied text gives no evidence that it has been resolved.