Restriction conjecture for the flat disk over finite fields

Let $n=2d\ge 4$ be even, let $\mathcal{F}$ be the flat disk in $\mathbb{F}_q^n$, and let $R^*_{\mathcal{F}}(p\to r)$ denote its restriction operator estimate. If $(1/p,1/r)$ lies in the convex hull of

$$(0,0),\quad (0,\tfrac{n-2}{2n}),\quad (\tfrac{n-2}{2n},\tfrac{n-2}{2n}),\quad (1,0),$$

then the flat-disk restriction conjecture.

$$R^*_{\mathcal{F}}(p\to r)\lesssim 1.$$

The convex-hull range is exactly the range suggested by the necessary conditions from the size of $\mathcal{F}$ and its affine subspaces. Sufficiency of these conditions is asserted here but is not established in the supplied text.