Critical endpoint conjecture for finite-field paraboloids

Let $q=|\mathbb F_q|$, and let $P\subset\mathbb F_q^d$ be the paraboloid

$$P=\{\xi\in\mathbb F_q^d:\xi_d=\xi_1^2+\cdots+\xi_{d-1}^2\}.$$

Write $R_P^*(p\to r)\lesssim1$ for the uniform finite-field extension estimate. Critical endpoint conjecture. The following estimates hold: if $d\geq2$ is even, then $R_P^*(\frac{2d^2}{d^2-d+2}\to\frac{2d}{d-1})\lesssim1$; if $d=4\ell-1$ for $\ell\in\mathbb N$ and $-1\in\mathbb F_q$ is not a square, then $R_P^*(\frac{2d^2+2d}{d^2+3}\to\frac{2d}{d-1})\lesssim1$; if $d=4\ell+1$ for $\ell\in\mathbb N$, then $R_P^*(\frac{2d}{d-1}\to\frac{2d}{d-1})\lesssim1$; and if $d\geq3$ is odd and $-1\in\mathbb F_q$ is a square, then $R_P^*(\frac{2d}{d-1}\to\frac{2d}{d-1})\lesssim1$. Establishing these endpoints would, together with the necessary-condition analysis, settle the extension problem for paraboloids.