Finite-field sphere extension conjecture

Let $\mathbb{F}_q^n$ be a finite-dimensional vector space, with $n\ge 2$ even, and let $S_r$ be the sphere centered at the origin with nonzero radius $r\in\mathbb{F}_q$. Denote by $R_{S_r}^*(2\to s)$ the normalized $L^2\to L^s$ extension constant for $S_r$.

Sphere extension conjecture. The $L^2\to L^{(2n+4)/n}$ extension estimate should hold uniformly in $q$:

$$R_{S_r}^*\left(2\to \frac{2n+4}{n}\right)\ll 1.$$

This is presented as the well-known finite-field $L^2\to L^r$ extension conjecture for spheres. It is used as an extension input for product-set estimates, and the supplied text gives no resolution status.