Distribuation of CM points of an infinite series of complete Calabi-Yau moduli spaces

In the infinite series of complete families of Calabi-Yau manifolds $\tilde{f}_n: \tilde{\mathcal{X}}_n\rightarrow \mathfrak{M}_{n, n+3}$, where $n$ is an odd number, arising from cyclic covers of $\mathbb{P}^n$ branching along hyperplane arrangements (\cite{SXZ13}), the set of CM points is dense for $n=1, 3$ and finite for $n\geq 5$.