The algebraic-closure comparison conjecture for ranks of subspaces

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, let $V_1,\ldots,V_d$ be $\mathbf k$-vector spaces, and let $L\subset (V_1\otimes\cdots\otimes V_d)^\vee$ be a linear subspace. Denote by $r_{\mathbf k}(L)$ its rank over $\mathbf k$ and by $r_{\bar{\mathbf k}}(L)$ its rank after extending scalars to $\bar{\mathbf k}$. The subspace-rank algebraic-closure conjecture. There exists a constant $E_d$ such that

$$r_{\mathbf k}(L)\leq E_d r_{\bar{\mathbf k}}(L).$$

This extends the preceding algebraic-closure comparison problem from a single multilinear polynomial to a linear subspace of multilinear forms. The source says that this conjecture is proved in a special case, but does not specify that case in the supplied text; the general statement remains open.