Unique nondegenerate minimum conjecture for the magnetic wedge edge eigenvalue

Let $\alpha_{0}\in(0,\pi)$, and let $\mu^\mathsf{e}(s,\eta)$ denote the lowest eigenvalue of the Neumann realization of

$$\mathcal{M}^{\mathsf{e}}_{s,\eta}=D_{t}^2+\mathcal{T}(s)^{-2}\mathcal{T}(0)^2D_{z}^2+(\eta-t)^2$$

on the sector $\mathcal{S}_{\alpha_{0}}$. Unique-minimum conjecture. The function $\eta\mapsto\mu^\mathsf{e}(0,\eta)$ admits a unique critical point $\eta_{0}$, which is a nondegenerate minimum. This conjecture concerns the spectral behavior of the magnetic Neumann Laplacian near a varying edge; its status is not determined by the supplied source.