Is Pi multiplied by Euler's Number Rational?

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

is rational or irrational.

Progress
By:
 * Transcendence of Sum or Product of Transcendentals
 * Euler's Number is Transcendental
 * Pi is Transcendental

at least one of $\pi + e$ and $\pi e$ is transcendental.

Also, see Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number.