Is Pi plus Euler's Number Rational?

Open Question
It is not known whether the sum of $\pi$(pi) and Euler's number $e$:
 * $\pi + e$

is rational or irrational.

Progress
If Schanuel's Conjecture is true, then $\pi + e$ is transcendental:

Let $z_1 = 1$ and $z_2 = -i \pi$.

By Schanuel's Conjecture, $\{1, -i \pi, e\}$ has transcendence degree at least $2$.

However, $1$ is algebraic.

Therefore, $-i \pi$ and $e$ must be algebraically independent.

Thus, $\pi$ and $e$ are algebraically independent.

And therefore $\pi + e$ is transcendental.