The Hölder upper-bound conjecture for metric mean dimension on the interval

Let $\phi:[0,1]\rightarrow [0,1]$ be an $\alpha$-Hölder continuous map. The metric mean dimension $\operatorname{mdim}_{\operatorname{M}}([0,1],|\cdot|,\phi)$ is defined using the interval with its Euclidean metric. Hölder upper-bound conjecture.

$$\operatorname{mdim}_{\operatorname{M}}([0,1],|\cdot|,\phi)\leq 1-\alpha.$$

The preceding theorem constructs examples attaining the bound for $\alpha\in(0,1)$, so the conjecture would give the optimal universal upper bound for Hölder maps on the interval. Its status is not resolved in the supplied text.