Nonexistence of positive-Hölder maps with full metric mean dimension

Let $\phi:[0,1]\rightarrow [0,1]$ be an $\alpha$-Hölder continuous map, with $\alpha>0$. The metric mean dimension $\operatorname{mdim}_{\operatorname{M}}([0,1],|\cdot|,\phi)$ is taken for the interval with its Euclidean metric. Nonexistence conjecture. There is no such map satisfying

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

The paper notes that this claim would follow from the preceding Hölder upper-bound conjecture, and relates it to the absence of continuous interval maps with metric mean dimension equal to $1$. Its status is not resolved in the supplied text.