Is Euler's Number to the power of Euler's Number Rational?

Open Question
It is not known whether Euler's number $e$ to the power of itself:
 * $e^e$

is rational or irrational.

Progress
If the Schanuel's Conjecture is true, then $e^e$ is transcendental:

Let $z_1 = 1$ and $z_2 = e$.

By Schanuel's Conjecture, the extension field $\Q \left({1, e, e, e^e}\right)$ has transcendence degree at least $2$ over $\Q$.

However, $1$ is algebraic.

Therefore, $e^e$ must be transcendental.