The sharp $L^2$ extension conjecture for paraboloids when $d=4k-1$

Let $P\subset\mathbb F_q^d$ be the paraboloid. Suppose that $d=4k-1$ for some $k\in\mathbb N$ and that $-1\in\mathbb F_q$ is not a square. Sharp paraboloid extension conjecture. One has

$$R^*_P\left(2\to\frac{2d+6}{d+1}\right)\ll 1,$$

which would give the sharp $L^2\to L^r$ extension estimate for $P$. This is identified as the difficult unresolved case following the known Stein–Tomas and partial even-dimensional results.