Stability conjecture for higher slice rank of subspaces

Let $d\geq 2$, let $\mathbb{K}$ be a field, let $\mathsf{W}\subseteq\mathbb{K}^{n_1}\otimes\cdots\otimes\mathbb{K}^{n_d}$ be a linear subspace, and let $k$ be a positive integer satisfying $k\leq\dim\mathsf{W}$. Write $\overline{\mathbb{K}}$ for an algebraic closure of $\mathbb{K}$. Stability conjecture for higher slice rank of subspaces. There is a constant $C\coloneqq C(d)>0$ such that

$$\operatorname{SR}_{k,\overline{\mathbb{K}}}(\mathsf{W}\otimes\overline{\mathbb{K}})\leq C\operatorname{SR}_{k,\mathbb{K}}(\mathsf{W}).$$

The source motivates this conjecture by the preceding results and says it is reasonable to expect stability under field extensions; it gives no resolution.