Non-convergence of ergodic measures with positive entropy

Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X$ with strictly positive Sinai entropy $h_{\mu}(T).$ Let $F:X\to X$ be a continuos transformation that commutes with $T.$

Define $F_{*}\mu(A)=\mu(F^{-1}A)$ for every Borel set $A.$

I am looking for an example in ergodic theory in which

$\frac{1}{N}\sum_{k=1}^NF_{*}^k\mu$ does not converge in the weak* topology as $N\to \infty.$

Pd: It is trivial to find examples in which $F_{*}^N\mu$ does not converge but $\frac{1}{N}\sum_{k=1}^NF_{*}^k\mu$ does.

Let $T_1, \hat{F} \colon X_1 \to X_1$ be commuting continuous transformations of a compact metric space and $\mu_1$ a Borel probability measure on $X_1$ which is invariant and ergodic with respect to $T_1$, and such that $\frac{1}{N}\sum_{k=1}^N {\hat{F}}_*^k\mu_1$ does not converge, as in your previous question. Let $T_2 \colon X_2 \to X_2$ be a continuous transformation of a compact metric space equipped with a $T_2$-invariant weak-mixing Borel probability measure $\mu_2$ such that $h(T_2,\mu_2)>0$. Define $X:=X_1 \times X_2$, $T:= T_1 \times T_2$, $F:=\hat{F} \times T_2$, $\mu:=\mu_1 \times \mu_2$.