<p>The purpose of this note is twofold. First, it aims to bring to light an apparently unnoticed property of the product of the extreme zeros of Laguerre polynomials, which in a very particular case leads to a twenty-year-old conjecture for Hermite polynomials posed by Gazeau, Josse-Michaux, and Monceau in the context of numerical methods in quantum mechanics. Second, it seeks to advance towards the resolution of this problem by means of a parametric eigenvalue approach.</p>