The monotonicity conjecture for crossing eigenvalue parameters

For each $n\in\mathbb N$, let $\beta_n$ be the unique parameter satisfying $\lambda(n,\beta_n)=\lambda(n+1,\beta_n)$, and define

$$\eta_n^*:=\eta(n,\beta_n)=\eta(n+1,\beta_n).$$

The crossing-sequence monotonicity conjecture. The sequence $(\eta_n^*)_{n\in\mathbb N}$ is strictly increasing. This conjecture is presented as a sufficient condition for the De Gennes upper-bound conjecture, linking the ordering of angular-momentum crossings to the global magnetic ground-state bound. Its resolution is not stated in the supplied excerpt.