Partition rank versus analytic rank conjecture

Let $d\geq 2$, let $q$ be a prime power, and let $T\in\mathbb{F}_q^{n_1}\otimes\cdots\otimes\mathbb{F}_q^{n_d}$, with $\operatorname{PR}_{\mathbb{F}_q}(T)$ and $\operatorname{AR}_{\mathbb{F}_q}(T)$ denoting its partition and analytic ranks. Partition rank vs. analytic rank conjecture. There exists a constant $C\coloneqq C(d,q)>0$ such that

$$\operatorname{PR}_{\mathbb{F}_q}(T)\leq C\operatorname{AR}_{\mathbb{F}_q}(T).$$

This is one of the paper’s three recalled rank-comparison conjectures; the source gives no resolution, while noting that versions with constants depending only on $d$ are the original formulations.