Katok's conjecture on entropy and periodic orbits for generic $C^1$ diffeomorphisms

Let $M$ be a compact finite-dimensional manifold and $f\colon M\to M$ be a diffeomrphism. By $P_n(f)$ we denote the number of periodic points of $f$ with period $n$, that is, the number of fixed points of $f^n$. Katok Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publications Mathématiques de l'IHÉS 51, no. 1 (1980), pp. 137-173 showed that the topological entropy of a $\mathcal{C}^{1+\varepsilon}$ diffeomorphism of a compact surface obeys the following inequality: $$\limsup_{n\to\infty} \frac{\log P_n(f)}{n}\ge h(f).$$ He also conjectured (see paragraph 5 page 141 of the article cited above) that this inequality holds in any dimension generically in the $\mathcal{C}^1$-topology. In other words: There is a dense $G_\delta$ set $\mathcal{G} \subset \text{Diff}^1 (M)$ such that for any $f\in\mathcal{G}$ one has $$\limsup_{n\to\infty} \frac{\log P_n(f)}{n}\ge h(f).$$ Is this conjecture still open? I would be grateful for any references.

The main point is that $C^1$-far from homoclinic tangencies the result must be true due to the existence of a maximal entropy measure (see here, here, here and references therein) where the splitting has one dimensional center (and so Katok type results can be carried out, see here). There is a subtelty here which is that the central exponent may be a priori equal to zero, in which case one would like to use the genericity hypothesis to show that the estimate holds (I think the concept of principal simbolic extension in the above references should be helpful for this). Here there might be some work to be done.