Hölder continuous maps on the interval with positive metric mean dimension

Fix a compact metric space X with finite topological dimension. Let C0(X) be the space of continuous maps on X and Hα(X) the space of α-Hölder continuous maps on X, for α ∈ (0, 1]. Let H1(X) be the space of Lipschitz continuous maps on X. We have H1(X) ⊂ Hβ(X) ⊂ Hα(X) ⊂ C0(X), where 0 < α < β...