The De Gennes upper-bound conjecture for the magnetic ground-state energy

Let $\beta>0$ denote the magnetic-field parameter, let $\lambda(\beta)$ be the lowest eigenvalue of the Neumann magnetic Laplacian in the unit disk, and define $\eta(\beta):=\lambda(\beta)/\beta$. Let $\Theta_0$ be the De Gennes constant, the infimum of the lowest eigenvalue $\lambda^{DG}(\xi)$ of the Neumann harmonic oscillator $D_t^2+(t+\xi)^2$ on $\mathbb R^+$ over $\xi\in\mathbb R$. The De Gennes upper-bound conjecture. For all $\beta>0$,

$$\eta(\beta)<\Theta_0.$$

This inequality is equivalent to $\lambda(\beta)<\Theta_0\beta$ and improves the general bound $\lambda(\beta)<\beta$. Since $\eta(\beta)\to\Theta_0$ as $\beta\to+\infty$, the conjecture concerns a strict finite-field upper bound and is not settled by the asymptotics.