Adiprasito–Schmidt stability conjecture for partition rank

Let $d\geq 2$, let $q$ be a prime power, and let $T\in\mathbb{F}_q^{n_1}\otimes\cdots\otimes\mathbb{F}_q^{n_d}$. Write $\overline{\mathbb{F}}_q$ for an algebraic closure of $\mathbb{F}_q$ and $T^{\overline{\mathbb{F}}_q}$ for the scalar extension of $T$. Stability conjecture for partition rank. There exists a constant $C\coloneqq C(d,q)>0$ such that

$$\operatorname{PR}_{\mathbb{F}_q}(T)\leq C\operatorname{PR}_{\overline{\mathbb{F}}_q}\left(T^{\overline{\mathbb{F}}_q}\right).$$

The source recalls this as an open stability conjecture; it also explains that the three rank conjectures recalled there are proved equivalent later in the paper.