We prove and substantially extend a conjecture of Gazeau, Josse-Michaux, and Monceau concerning the extreme positive zeros of Hermite polynomials, which arose from a finite-dimensional quantisation of the phase plane. For the generalised Laguerre polynomial L(α)m of degree m and parameter α>−1, the product of its jth smallest and jth largest zeros is strictly increasing with m for each fixed positive integer j, once m≥2j−1. The classical Hermite-Laguerre identities then prove the original conjecture for the degree-N Hermite polynomial HN, in both parities and for every N≥4. As a further consequence, a one-parameter extension involving a deformation of the corresponding Jacobi matrix is settled.