IMC 2000 · Problem 6

For an $m \times m$ real matrix $A$, $e^A$ is defined as $\sum\limits_{n=0}^{\infty} \dfrac{1}{n!} A^n$. (The sum is convergent for all matrices.) Prove or disprove, that for all real polynomials $p$ and $m \times m$ real matrices $A$ and $B$, $p(e^{AB})$ is nilpotent if and only if $p(e^{BA})$ is nilpotent. (A matrix $A$ is nilpotent if $A^k = 0$ for some positive integer $k$.)