Local variational principle for partitions of unity

Establish that for every continuous partition of unity Φ on a compact metric space X and continuous self-map T: X → X, the supremum over all T-invariant Borel probability measures of the local metric entropy defined via continuous partitions of unity equals the local topological entropy defined via continuous partitions of unity; that is, determine whether sup_{μ ∈ M(X,T)} h_μ(T, Φ) = h(T, Φ).

Background

The paper introduces local metric entropy h_μ(T, Φ) and local topological entropy h(T, Φ) defined using continuous partitions of unity Φ, and proves properties including upper semi-continuity of μ ↦ h_μ(T, Φ) and attainment of the supremum.

In the setting of open covers, Romagnoli previously proved a local variational principle equating the supremum of local metric entropy with local topological entropy. The authors ask whether an analogous equality holds for their partition-of-unity based definitions.