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.