I posted this question on mathoverflow. The day after I posted it, I figured out that it seems to be equivalent to the following:
You have a doubly periodic holomorphic function with periods $2\pi$ and $2\pi i$ that takes $0$ and $\pi + i\pi$ to $0$ and takes $\pi$ and $i\pi$ to $\infty.$
Now look at the inverse-images of straight lines through $0$ corresponding to arguments (i.e. angles from the real axis) between $0$ and $45^\circ$. Those inverse-images should be graphs of functions of the form $y=\arcsin(c\sin(x))$ where $x$ and $y$ are respectively the real and imaginary parts.
Is that a known proposition about Jacobi's functions?
Now that I've reduced it to this seemingly more mundane form, is it now something that everybody knows? (Except those who don't know the theory of Jacobi's functions?)