Let $X=\{1,\dots,p\}^\mathbb{N}$ be the space of sequences on a finite alphabet with a metric inducing the product topology, and let $\sigma\colon X\to X$ be the shift map. Let $\mu$ be a $\sigma$-invariant Borel probability measure on $X$.
A point $x\in X$ is generic for $\mu$ if $\frac 1n S_n\phi(x) \to \int \phi\\,d\mu$ for every $\phi\in C(X)$, where $S_n \phi(x) = \phi(x) + \phi(\sigma x) + \cdots + \phi(\sigma^{n-1}x)$. Denote by $G_\mu$ the set of $\mu$-generic points.
Fact #1. If $\mu$ is ergodic, then Birkhoff's ergodic theorem implies that $\mu(G_\mu)=1$.
The local entropy of a point $x\in X$ is $h_\mu(x) = \lim_{n\to\infty} -\frac 1n \log \mu([x_1\dots x_n])$, where $[x_1 \dots x_n] = \{ y\in X \mid y_i = x_i \\,\forall 1\leq i\leq n\}$, provided the limit exists. Denote by $Z_\mu$ the set of points $x$ for which $h_\mu(x)$ exists and is equal to the measure-theoretic entropy $h_\mu(\sigma)$.
Fact #2. If $\mu$ is ergodic, then Shannon-McMillan-Breiman implies that $\mu(Z_\mu)=1$.
The measure $\mu$ is a Gibbs measure if there exists a function $\phi\in C(X)$ and constants $K,P>0$ such that $$ K^{-1} \leq \frac{\mu([x_1\dots x_n])}{e^{-nP + S_n \phi(x)}} \leq K $$ for every $x\in X$ and $n\in \mathbb{N}$.
Fact #3. If $\mu$ is a Gibbs measure, then $G_\mu \subset Z_\mu$. That is, the local entropy of a point $x$ with respect to $\mu$ is "what it should be" provided the Birkhoff averages of continuous functions along the orbit of $x$ are "what they should be".
(Actually, even more is true: for a Gibbs measure the local entropy $h_\mu(x)$ of any point $x$ is completely determined by the Birkhoff averages $\frac 1n S_n \phi(x)$ of a single function.)
Question. What is the broadest class of measures for which the inclusion $G_\mu \subset Z_\mu$ holds -- that is, for which genericity for Birkhoff averages of continuous functions implies genericity for local entropies? Does this hold for all ergodic measures? If it does not, is there a natural class of measures beyond the Gibbs measures (and various notions of weak Gibbs measures) for which it does hold?
Related question. Gibbs measures (and weak Gibbs measures) have the property that there exists a function $\phi\in C(X)$ such that $h_\mu(x)$ is completely determined by $\frac 1n S_n \phi(x)$ for every $x\in X$. (Not just for a full measure set of $x$ -- this is true for all ergodic measures.) Is there an example of a measure $\mu$ such that there is no single function $\phi\in C(X)$ whose Birkhoff averages determine $h_\mu(x)$, but there exist two function $\phi_1, \phi_2\in C(X)$ such that $h_\mu(x)$ is completely determined by $\frac 1n S_n \phi_i(x)$ for $i=1,2$?