The algebraic-closure comparison conjecture for multilinear rank

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, and let $P$ be a multilinear $\mathbf k$-polynomial of degree $d\geq 2$. Write $r_{\mathbf k}(P)$ for its rank and $r_{\bar{\mathbf k}}(P)$ for the rank after extending scalars to $\bar{\mathbf k}$. The algebraic-closure comparison conjecture. For every $d\geq 2$, there exists $\kappa_d>0$ such that

$$r_{\mathbf k}(P)\leq \kappa_d r_{\bar{\mathbf k}}(P).$$

The reverse inequality is immediate, and the conjecture asks for a bound in the other direction independent of the field and polynomial. It is known for $d=2$ with $\kappa_2=1$ and, for trilinear polynomials, follows with constant $3/2$ from the theorem proved in the paper; the general case remains open.