Schanuel's conjecture is an important conjecture in transcendental number theory, which is:

Given any $n$ complex numbers $z_1, z_2, ..., z_n$ that are linearly independent over the rational numbers $\Bbb Q$, the field extension $\Bbb Q(z_1, ... z_n, e^{z_1}, ..., e^{z_n})$ has transcendence degree at least $n$ over $\Bbb Q$.

Wikipedia lists some consequences of this being true. It would strengthen both the Lindemann–Weierstrass theorem and Baker's theorem by showing that linearly independent logarithms of algebraic numbers are also algebraically independent, which would in turn make it easy to settle the transcendence of $\pi+e$, $e^e$, and so on, as well as that $e$ and $\pi$ are algebraically independent.

**My question**: what would be some of the results if this conjecture were instead false? How would that change transcendental number theory?