The Montgomery family quartic oscillator eigenvalue critical-point conjecture

Let $\lambda_j(\alpha)$ denote the $j$-th eigenvalue of the Montgomery family of quartic oscillators, for $j\geq 1$ and $\alpha\in\mathbb R$. A critical point $\alpha_{j,c}$ satisfies $\lambda_j'(\alpha_{j,c})=0$. Montgomery eigenvalue critical-point conjecture. For any $j\geq 1$, $\lambda_j$ has a unique critical point $\alpha_{j,c}$. Moreover, this critical point is positive, corresponds to a minimum, and is non-degenerate. The conjecture unifies the proved result for the first eigenvalue, the numerically assisted result for $\lambda_2$, and the theorem establishing the corresponding statement for all sufficiently large eigenvalues; the intermediate eigenvalues remain to be settled.