Finite historic-measure mass implies zero entropy

Let ${\mathbb X}$ be a compact manifold and let $f:{\mathbb X}\to{\mathbb X}$ be a smooth ($C^{1+}$) diffeomorphism. For each $x\in{\mathbb X}$, let $\eta_x$ denote the associated historic measure.

Zero-entropy conjecture. If

$$\eta_x({\mathbb X})<\infty$$

for every $x\in{\mathbb X}$, then $f$ has zero topological entropy:

$$h_{\mathrm{top}}(f)=0.$$

The claim is disproved for homeomorphisms, by examples of Bégin, Crovisier and Le Roux, but remains conjectural in the stated smooth diffeomorphism setting.