The sharp even-dimensional sphere extension conjecture

Let $d\ge 2$ be even, and let $S_j\subset\mathbb F_q^d$ be the sphere of radius $j\ne 0$ centered at the origin. Even-dimensional sphere extension conjecture. One has

$$R^*_{S_j}\left(2\to\frac{2d+4}{d}\right)\ll 1.$$

This estimate is expected to be sharp for the $L^2\to L^r$ extension problem. The paper presents it as the even-dimensional counterpart to the odd-dimensional sphere conjectures, beyond the Stein–Tomas estimate.