I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is possible is $\Sigma^0_2$, that is, we do not know if there is a minimal tds with an invariant measure such that the set of generic points for that measure is a countable union of closed sets (it is an $F_\sigma$ set) that is not a countable intersection of open sets (it is not a $G_\delta$ set).
Notation & definitions
A topological dynamical system (tds) is a compact metric space $X$ together with a continuous map $T\colon X\to X$. The orbit of a point $x\in X$ in a tds $(X,T)$ is the set $\{T^n(x)\mid n\ge 0\}$. We say that a tds $(X,T)$ is minimal if $X$ is the only nonempty and closed set $A\subseteq X$ satisfying $T(A)\subseteq A$, equivalently all orbits are dense. We say that a tds $(X,T)$ is transitive if $T$ is surjective and there exists a point in $X$ whose orbit is dense. Let $\mu$ be a Borel $T$-invariant (meaning $T_*\mu=\mu$) probability measure on $X$. A point $x\in X$ is generic for $\mu$ if for every continuous real-valued function $f$ on $X$ we have $$\lim_{N\to\infty}\frac1N\sum_{n=0}^{N-1}f(T^j(x))=\int_X f\;\text{d}\mu.$$
What is known?
(these are both our results with Konrad Deka and what we have found in the literature)
Given a $T$-invariant measure $\mu$, the set $\text{Gen}(\mu)$ of points generic for $\mu$ can always be presented as an countable intersection of countable unions of closed sets (it is $F_{\sigma\delta}$ or $\Pi_3^0$), hence $\text{Gen}(\mu)$ is always a Borel set. We know examples of a minimal tds $(X,T)$ such that
there is a $T$-invariant measure $\mu$ such that $\text{Gen}(\mu)$ cannot be presented as an countable union of countable intersections of open sets (it is not $G_{\delta\sigma}$ or $\Sigma^0_3$, so it is $\Pi^0_3$-complete),
there is a $T$-invariant measure $\mu$ such that $\text{Gen}(\mu)$ can be presented as an countable union of countable intersections of open sets (it is $\Sigma^0_3$ and $\Pi^0_3$, so it is $\Delta^0_3$), but neither $G_\delta$ nor $F_\sigma$,
if $\mu$ is a $T$-invariant measure such that $\text{Gen}(\mu)$ is $G_\delta$, then $\text{Gen}(\mu)=X$, so $T$ is uniquely ergodic.
We also know that there exists a transitive tds $(X,T)$ that has an invariant measure $\mu$ such that $\text{Gen}(\mu)$ is $F_\sigma$ but not $G_\delta$.
Question
Is there a minimal tds $(X,T)$ that has an invariant measure $\mu$
such that $\text{Gen}(\mu)$ is $F_\sigma$ but not $G_\delta$?
