Adiprasito–Kazhdan–Ziegler partition-rank versus analytic-rank conjecture

Let $\mathbb{K}$ be a finite 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 $\operatorname{PR}_{\mathbb{K}}(f)$ is the partition rank and $\operatorname{AR}_{\mathbb{K}}(f)$ is the analytic rank of $f$. Adiprasito–Kazhdan–Ziegler's partition-rank versus analytic-rank conjecture. One has

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

The relation $\asymp_d$ denotes linear equivalence with constants depending on $d$. The conjecture is known up to a logarithmic factor, but the exact linear equivalence remains open according to the source.