Paraboloid $L^2$ extension range conjecture

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 $R_P^*(p\to r)\lesssim 1$ when the finite-field extension operator for $P$ is bounded from $L^p(P)$ to $L^r(\mathbb F_q^d)$ uniformly in $q$. The $L^2$ extension range conjecture. The following equivalences hold: if $d\geq 2$ is even, then $R_P^*(2\to r)\lesssim 1$ if and only if $\frac{2d+4}{d}\leq r\leq\infty$; if $d=4\ell-1$ for $\ell\in\mathbb N$ and $-1\in\mathbb F_q$ is not a square, then $R_P^*(2\to r)\lesssim 1$ if and only if $\frac{2d+6}{d+1}\leq r\leq\infty$; if $d=4\ell+1$ for $\ell\in\mathbb N$, then $R_P^*(2\to r)\lesssim 1$ if and only if $\frac{2d+2}{d-1}\leq r\leq\infty$; and if $d\geq 3$ is odd and $-1\in\mathbb F_q$ is a square, then $R_P^*(2\to r)\lesssim 1$ if and only if $\frac{2d+2}{d-1}\leq r\leq\infty$. The reverse implications follow from the paper's necessary-condition lemma, while the forward implications are known in dimension two but remain open in higher even dimensions.