Comment on "Which is greater: $e^{\pi }$ or ${\pi }^e$? An unorthodox physical solution to a classic puzzle"

In a recent Note (Am. J. Phys. 92:397, 2024; arXiv:2309.10826), Vallejo and Bove provide a physical argument based nominally on the second law of thermodynamics as a way of resolving the mathematical question appearing in the title. A remarkable aspect of their argument is that it does not depend on the numerical value of $\pi$, because $e^x\ge x^e$ for all positive $x$, with equality occurring only when $x=e$. Moreover, their argument does not depend on the validity of the second law but is rather a limited proof of it for this particular case.