Vanishing metric mean dimension for Hölder maps on the interval

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

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

This claim is stronger than the preceding upper-bound conjecture and is presented as a further conjecture about the relationship between Hölder regularity and metric mean dimension. No resolution is given in the supplied text.