Uniqueness conjecture for the minimizer of the ground-state energy

Let $\lambda_0(\alpha)$ be the ground state energy of the operator defined in the paper, and let $\hat{\nu}$ be its minimum value. A minimizer is a point $\alpha_{\rm \min}\in\mathbb{R}$ satisfying $\lambda_0(\alpha_{\rm \min})=\hat{\nu}$. Uniqueness conjecture. There exists a unique $\alpha_{\rm \min}\in\mathbb{R}$ such that

$$\lambda_0(\alpha_{\rm \min})=\hat{\nu}.$$

The source presents this as an additional condition intended to yield a lower bound on the splitting between the two lowest eigenvalues of the magnetic Schrödinger operator. No resolution is stated.