The powers-of-singular-moduli linear equation conjecture

Let $A,B,C$ be rational numbers with $AB\ne 0$, let $x,y$ be singular moduli, and let $m,n$ be positive integers. Assume that

$$Ax^m+By^n=C.$$

Powers-of-singular-moduli linear equation conjecture. Then one of the following holds: $A+B=C=0$, $x=y$, and $m=n$; $x,y\in\mathbb Q$; or $x\ne y$ and $\mathbb Q(x)=\mathbb Q(y)$ is a number field of degree $2$. The conjecture generalizes the cited classifications for linear equations in singular moduli and for products of two singular moduli by allowing positive integer exponents. The source gives no resolution, so the conjecture remains open.