Invariant-measure variational principle for r-neutralized topological entropy

Let $(X,d)$ be a compact metric space, let $f:X\to X$ be a homeomorphism, and let $\mathcal{M}(f,X)$ be the invariant Borel probability measures for $f$. For $r>0$, define the $r$-neutralized topological entropy and lower $r$-neutralized topological entropy by

$$h_{d}^r(f):=\varlimsup_{n \to \infty} \frac{1}{n}\log S_{d}(f,n,e^{-nr}),\qquad \underline{h}_{d}^r(f):=\varliminf_{n \to \infty}\frac{1}{n}\log S_{d}(f,n,e^{-nr}).$$

Here $S_d(f,n,e^{-nr})$ is the minimal cardinality of an $(n,e^{-nr})$-spanning set. Invariant-measure variational principle. For every $r>0$,

$$h_{d}^r(f)=\sup\{h_{\nu,d}^r(f):\mu\in\mathcal{M}(f,X)\},\qquad \underline{h}_{d}^r(f)=\sup\{\underline{h}_{\nu,d}^r(f):\mu\in\mathcal{M}(f,X)\}.$$

This is proposed as an analogue of the classical variational principle, replacing the supremum over Borel probability measures by one over invariant measures. The claim is stated for homeomorphisms and both upper and lower $r$-neutralized entropies; the source provides no resolution.