Let $S$ denote the closed oriented surface of genus $g\geq 2$, and $\text{Mod}(S)$ be the mapping class group of $S$. Let $f\in \text{Mod}(S)$ be a finite order reducible element i.e. $f$ has a representative $\phi\in \text{Homeo}^+(S)$ such that $\phi^k=$ identity for some $k>1$ and $\phi$ preserve a multi-curve in $S$. There is a natural action of the cyclic group $\text{Mod}(S)$ on the Thurston boundary $\mathcal{PMF}(S)$. Is the set $$\text{Fix}(f):=\{[(\mathcal{F},\mu)]\in \mathcal{PMF}(S)|\, f.[(\mathcal{F},\mu)]=[(\mathcal{F},\mu)]\},$$ always a finite set?