Extended asymptotic rank conjecture for concise tensors

Let $\mathbb{F}$ be the underlying field, and consider concise tensors in

$$\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d.$$

Write $\sigma(d)$ for the worst-case exponent of this space. Extended asymptotic rank conjecture. For all $d\geq 1$ it holds that

$$\sigma(d)=1.$$

This is stronger than Strassen's asymptotic rank conjecture and asserts that the least possible exponent is shared by all concise tensors.