Asymptotic direct sum conjecture for partition rank

Let $d\geq 2$, let $q$ be a prime power, and let $T\in\mathbb{F}_q^{n_1}\otimes\cdots\otimes\mathbb{F}_q^{n_d}$. For $n\geq 1$, let $T^{\oplus n}$ denote the $n$-fold direct sum of $T$. Asymptotic direct sum conjecture for partition rank. There exists a constant $C\coloneqq C(d,q)>0$ such that

$$\operatorname{PR}_{\mathbb{F}_q}(T)\leq C\limsup_{n\to\infty}\frac{\operatorname{PR}_{\mathbb{F}_q}(T^{\oplus n})}{n}.$$

The paper presents this as another conjectural formulation and later states that it is equivalent to the preceding partition-rank conjectures; no resolution of the common conjecture is supplied.