How far is the slice rank of a tensor from its CP rank

Question

Assume we work on any infinite field and 3-ordered tensor. Clearly for any tensor $T$, we have $\operatorname{srk}(T)\le \operatorname{rk}(T)$. Here, $\operatorname{srk}(T)$ (resp. $\operatorname{rk}(T)$) denotes the slice rank (resp. CP rank) of $T$ My question is if there is a constant $c>0$ which does not depending on $T$ such that $\operatorname{rk}(T)\le c\cdot \operatorname{srk}(T)$.

Answer 1

No.

For $n \times n \times n$ tensors slice rank does not exceed $n$, while tensor rank can be as large as $\frac{n^3}{3n - 2} \sim \frac{n^2}{3}$ by dimension count.