The constant-factor equivalence conjecture for partition and analytic rank

Let $k$-tensors be tensors $T$ over a finite field $F$, with partition rank $\operatorname{PR}(T)$ and analytic rank $\operatorname{AR}(T)$.

Constant-factor equivalence conjecture. For every $k ge 2$ there is a constant $C=C(k)$ such that, for every finite field $F$ and every $k$-tensor $T$ over $F$,

$$\operatorname{PR}(T) \le C\cdot\operatorname{AR}(T).$$

The conjecture asserts that the structure measured by partition rank and the randomness measured by analytic rank are equivalent up to a constant factor. The inequality $\operatorname{AR}(T)\le\operatorname{PR}(T)$ is known, and polynomial upper bounds for partition rank in terms of analytic rank are available, but this uniform linear bound remains open in the stated generality.