  Pila and Tsimerman proved in 2017 that for every $k$ there exists at most
finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular
moduli with the property "$x_1, \ldots,x_k$ are multiplicatively dependent, but
any proper subset of them is multiplicatively independent". The proof was
non-effective, using Siegel's lower bound for the Class Number. In 2019 Riffaut
obtained an effective version of this result for $k=2$. Moreover, he determined
all the instances of $x^my^n\in \mathbb Q^\times$, where $x,y$ are distinct
singular moduli and $m,n$ non-zero integers. In this article we obtain a
similar result for $k=3$. We show that $x^my^nz^r\in \mathbb Q^\times$ (where
$x,y,z$ are distinct singular moduli and $m,n,r$ non-zero integers) implies
that the discriminants of $x,y,z$ do not exceed $10^{10}$.
