Sharp persistence-exponent conjecture for Hölder functions

Let $X$) be a compact Riemannian manifold of dimension $d$, and let $f\in C^{r+\alpha}(X)$, with the persistence quantity $\Pers_p(H_k(X,f))$ defined as the $p$-persistence of the degree-$k$ persistent homology of the sublevel-set filtration of $f$. Sharpness conjecture.

$$\inf\{p \mid \Pers_p(H_k(X,f))<\infty\}\leq\frac{d}{r+\alpha}$$

Moreover, this inequality is saturated generically in the sense of Baire in $C^{r+\alpha}$. The conjecture proposes that the finiteness threshold obtained from regularity estimates is optimal, extending the known generic sharpness result for $\alpha$-Hölder functions; it is presented as a question for future work, so its status is open.