Multiplicative relations among differences of singular moduli

Let $n \in \mathbb{Z}_{>0}$. We prove that there exist a finite set $V$ and finitely many algebraic curves $T_1, \ldots, T_k$ with the following property: if $(x_1, \ldots, x_n, y)$ is an $(n+1)$-tuple of pairwise distinct singular moduli such that $\prod_{i=1}^n (x_i - y)^{a_i}=1$ for some $a_1, \ldots, a_n \in \mathbb{Z} \setminus \{0\}$, then $(x_1, \ldots, x_n, y) \in V \cup T_1 \cup \ldots \cup T_k$. Further, the curves $T_1, \ldots, T_k$ may be determined explicitly for a given $n$.