A little reading suggests:

It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is transcendental.

If we just want irrationality rather than transcendence, is a proof known?

Can we prove $\pi+e$ is irrational? Can we prove $\pi \times e$ is irrational?