On the Hofer-Zehnder conjecture for semipositive symplectic manifolds

We show that, on a closed semipositive symplectic manifold with semisimple quantum homology, any Hamiltonian diffeomorphism possessing more contractible fixed points, counted homologically, than the total Betti number of the manifold, must have infinitely many periodic points. This generalizes to the semipositive setting the beautiful result of Shelukhin on the Hofer-Zehnder conjecture.