Is there any $p \in \mathbb{Q}[x,y]$ with x and y degree both at least 1 such that $p(\pi,e)$ is known to be irrational?

Question

The algebraic independence of $\pi$ and $e$ is a well known open problem, as is the specific case of rationality of $\pi + e$. My question is if there is any polynomial with rational coefficients (other than the obvious ones where e.g. $p(x,y)=q(x)$) such that $p(\pi,e)$ has been proven irrational.