Conjecture on rank comparison over a field and its algebraic closure

Let $K$ be a field, let $V=V_1\times\dots\times V_d$, and let $P:V\to K$ be a multilinear polynomial of degree $d$. Write $r_K(P)$ for the Schmidt rank of $P$, and let $\bar K$ be the algebraic closure of $K$. For any $d\geq 2$, rank-comparison conjecture. there exists $\kappa_d>0$ such that

$$r_K(P)\leq\kappa_d r_{\bar K}(P).$$

The claim is known for $d=2$, with $\kappa_2=1/2$, and for $d=3$ because Schmidt rank equals slice rank and the required comparison follows from the cited results. The general statement remains open.