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 $T_1,\dots ,T_k$ with the following property: if $(x_1,\dots ,x_n,y)$ is an $(n+1)$-tuple of pairwise distinct singular moduli such that ${\prod }_{i=1}^n(x_i-y{)}^{a_i}=1$ for some $a_1,\dots ,a_n\in \mathbb{Z}\setminus \{0\}$, then $(x_1,\dots ,x_n,y)\in V\cup T_1\cup \dots \cup T_k$. Further, the curves $T_1,\dots ,T_k$ may be determined explicitly for a given $n$.