Effective multiplicative independence of three singular moduli

Pila and Tsimerman proved in 2017 that for every k there exists at most finitely many k-tuples (x₁,…, xₖ) of distinct non-zero singular moduli with the property"x₁, …,xₖ 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 xᵐyⁿ∈ Q^×, 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 xᵐyⁿzʳ∈ Q^× (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 10¹⁰.