The sharp $L^p\to L^4$ extension conjecture for odd-dimensional spheres

Let $S_j\subset\mathbb F_q^d$ be a sphere of nonzero radius $j$. Odd-dimensional sphere extension conjecture. The following estimates should hold:

1. If $d=4k+1$, $k\in\mathbb N$, and $j$ is nonsquare, then

$$R^*_{S_j}\left(\frac{4d+4}{3d+1}\to 4\right)\ll 1.$$

2. If $d=4k-1$, $k\in\mathbb N$, $q\equiv 1\pmod 4$, and $j$ is nonsquare, the same estimate holds.

3. If $d=4k-1$, $k\in\mathbb N$, $q\equiv 3\pmod 4$, and $j$ is square, the same estimate holds.

In each case the estimate is expected to be sharp for the $L^p\to L^4$ extension problem. The conjecture is motivated by the affine-subspace dimensions on spheres and is compared with known paraboloid results; only partial evidence is given.