Adiprasito–Kazhdan–Ziegler stability conjecture for partition rank

Let $\mathbb{K}$ be a field, let $d\geq 2$ be an integer, and let

$$f\in\operatorname{Hom}(\mathbb{K}^{n_1}\times\cdots\times\mathbb{K}^{n_d},\mathbb{K}).$$

Here $\operatorname{PR}_{\mathbb{K}}(f)$ denotes the partition rank of $f$, and $\overline{\mathbb{K}}$ is an algebraic closure of $\mathbb{K}$. Adiprasito–Kazhdan–Ziegler's stability conjecture. One has

$$\operatorname{PR}_{\mathbb{K}}(f)\asymp_d\operatorname{PR}_{\overline{\mathbb{K}}}(f).$$

The relation $\asymp_d$ denotes linear equivalence with constants depending on $d$. The conjecture is now proved for perfect infinite fields by the theorem stated in the source, but its status in the full generality of arbitrary fields is not resolved there.