Winkelmann's upper-half-plane formulation of the isogeny criterion

Let $H^+$ be the upper half-plane, and define

$$B^+(\mathbb Q)=\left\{\begin{pmatrix}a&b\\0&a^{-1}\end{pmatrix}:a\in\mathbb Q^+,\ b\in\mathbb Q\right\}.$$

The groups $GL_2^+(\mathbb Q)$ and $B^+(\mathbb Q)$ act on $H^+$ by fractional linear transformations. Winkelmann's upper-half-plane conjecture. If $\sigma,\tau\in H^+$ lie in the same $GL_2^+(\mathbb Q)$-orbit and both $e^{2\pi i\sigma}$ and $e^{2\pi i\tau}$ are algebraic, then $\sigma$ and $\tau$ lie in the same $B^+(\mathbb Q)$-orbit. The source presents this as an equivalent reformulation of the preceding isogeny criterion and proves the equivalent conjectures assuming Schanuel's conjecture.