Winkelmann's isogeny criterion for algebraic multiplicative quotients

Let $\alpha_1,\alpha_2\in\mathbb C$ be algebraic numbers with

$$|\alpha_i|>1.$$

Let

$$E_i=\mathbb C^*/\{\alpha_i^k:k\in\mathbb Z\}.$$

Two nonzero complex numbers are multiplicatively dependent if there are integers $(m,n)\ne(0,0)$ such that $\alpha_1^m\alpha_2^n=1$. Winkelmann's isogeny criterion. The curves $E_1$ and $E_2$ are isogenous if and only if $\alpha_1$ and $\alpha_2$ are multiplicatively dependent. The paper states this criterion as a conjecture and explains that it is equivalent to a formulation involving $GL_2^+(\mathbb Q)$- and $B^+(\mathbb Q)$-orbits; it proves the criterion assuming Schanuel's conjecture.