The Schmidt rank versus analytic rank conjecture for multilinear polynomials

Let $d\geq 2$, let $\mathbb F_q$ be a finite field, and let $P$ be a multilinear polynomial of degree $d$. Denote by $r_{\mathbb F_q}(P)$ its Schmidt rank and by $a_{\mathbb F_q}(P)$ its analytic rank. The Schmidt-rank versus analytic-rank conjecture. There exists $c_d>0$ such that

$$r_{\mathbb F_q}(P)\leq c_d a_{\mathbb F_q}(P).$$

The inequality in the opposite direction is known, and the conjecture is known for $d=2$ with $c_2=1$ and for trilinear polynomials with $c_3=3$. Earlier work established a polynomial bound in the analytic rank, while this conjecture asks for a linear bound; the general case remains open.