The uniqueness and non-degeneracy conjecture for the lowest band-function minimum

Let $\zeta_{1}(\tau)$ be the lowest band function associated with the three-dimensional magnetic Hamiltonian with axisymmetric potential. The preceding analysis shows that $\zeta_{1}$ is non-increasing on $(-\infty,0)$, and numerical simulations suggest that $\zeta_{2}(\tau)-3\zeta_{1}(\tau)>0$ for $\tau>0$. Uniqueness and non-degeneracy conjecture. The band function $\tau\mapsto\zeta_{1}(\tau)$ has a unique and non-degenerate minimum. This conjecture would determine the structure of the lowest band function and its critical points, building on the known monotonicity for negative $\tau$ and the observed positivity of the spectral gap; its resolution is not established here.