A max inequality for spectral invariants of disjointly supported Hamiltonians

We study the relation between spectral invariants of disjointly supported Hamiltonians and of their sum. On aspherical manifolds, such a relation was established by Humilière, Le Roux and Seyfaddini. We show that a weaker statement holds in a wider setting, and derive applications to Polterovich's Poisson bracket invariant and to Entov and Polterovich's notion of superheavy sets.