Conjecture on analytic rank and Schmidt rank over finite fields

Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements, and let $P$ be a multilinear polynomial of degree $d$. Write $a_{\mathbb F_q}(P)$ for its analytic rank and $r_{\mathbb F_q}(P)$ for its Schmidt rank. Analytic-rank comparison conjecture. For any $d\geq 2$, there exists $\epsilon_d>0$ such that

$$a_{\mathbb F_q}(P)\geq\epsilon_d r_{\mathbb F_q}(P).$$

This asks for a uniform linear lower bound relating analytic rank and Schmidt rank over finite fields. The supplied excerpt gives no resolution or partial results for this conjecture, so its general status remains open.