Suppose $\ \pi+e\ $ is a transcendental number.
Can we conclude whether $\ \pi-e\ $ is rational, algebraic irrational or transcendental ?
If I understood the consequences of Schanuel's conjecture correctly, it implies that $\ \pi-e\ $ is trancendental. But can we also say something without any further unproven conjecture ?
I claim that that statement "if $\pi+e$ is transcendental, then $\pi-e$ is transcendental" implies that $\pi-e$ is transcendental. This shows that the knowledge that $\pi+e$ is transcendental is of no help in concluding that $\pi-e$ is transcendental.
Assume that if $\pi+e$ is transcendental, then $\pi-e$ is transcendental. Separately, we can see directly that if $\pi+e$ is algebraic, then $\pi-e = (\pi+e)-2e$ is transcendental (since we know $e$ is transcendental). Since $\pi+e$ is either transcendental or algebraic, we conclude that $\pi-e$ is transcendental.