Effective multiplicative independence of 3 singular moduli

Abstract: 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 $10^{10}$.