Equidistribution conjecture for unstable periodic points of heterochaos baker maps

Let $(a,b)\in\Delta$, let $f$ be the corresponding heterochaos baker map on $[0,1]^3$, and let $\mu_\alpha$ and $\mu_\beta$ be the two ergodic measures of maximal entropy. Denote by $\operatorname{Fix}_\alpha(f^n)$ and $\operatorname{Fix}_\beta(f^n)$ the sets of $\alpha$- and $\beta$-type fixed points of $f^n$. For every continuous function $\phi\colon[0,1]^3\to\mathbb R$, Equidistribution conjecture. For any $(a,b)\in\Delta$ and any continuous function $\phi\colon[0,1]^3\to\mathbb R$,

$$\lim_{n\to\infty}\frac{1}{\#\operatorname{Fix}_{\alpha}(f^n)}\sum_{x\in\operatorname{Fix}_{\alpha}(f^n)}\phi(x)=\int\phi\,\mathrm d\mu_{\alpha}$$

and

$$\lim_{n\to\infty}\frac{1}{\#\operatorname{Fix}_{\beta}(f^n)}\sum_{x\in\operatorname{Fix}_{\beta}(f^n)}\phi(x)=\int\phi\,\mathrm d\mu_{\beta}.$$

This predicts that the two unstable periodic-point populations equidistribute toward the two ergodic measures of maximal entropy.