Shelukhin's Hofer-Zehnder conjecture for local Floer homology

Let $(M,\omega)$ be a closed symplectic manifold and let ${\mathbb K}$ be a ground field. For a possibly degenerate Hamiltonian diffeomorphism $\phi$, define

$$N(\phi,{\mathbb K})=\sum_{x\in\operatorname{Fix}(\phi)}\dim_{\mathbb K}\operatorname{HF}^{\operatorname{loc}}(\phi,x).$$

Shelukhin's Hofer-Zehnder conjecture. If

$$N(\phi,\mathbb K)>\dim_{\mathbb K}\operatorname{H}_{*}(M;\mathbb K),$$

then $\phi$ must have infinitely many periodic points. This is a precise Floer-theoretic interpretation of the Hofer-Zehnder conjecture that includes degenerate Hamiltonian diffeomorphisms. The source discusses the conjecture as part of the ongoing effort to prove the Hofer-Zehnder statement beyond the symplectically aspherical and spherically monotone cases.