Non-degeneracy conjecture for the minima of the ground-state eigenvalue

Let $k$ be a positive integer, let $Q(\alpha,1)$ be the operator whose lowest eigenvalue is $\lambda_0(\alpha,1)$, and let $\hat{\nu}$ denote the minimum value of $\lambda_0(\alpha,1)$. Thus, $\alpha_{\rm \min}\in\mathbb{R}$ is a minimizer when $\lambda_0(\alpha_{\rm \min},1)=\hat{\nu}$. Non-degeneracy conjecture. Any minimum of $\lambda_0(\alpha,1)$ is non-degenerate: for every $\alpha_{\rm \min}\in\mathbb{R}$ such that $\lambda_0(\alpha_{\rm \min},1)=\hat{\nu}$,

$$\frac{\partial^2\lambda_0}{\partial\alpha^2}(\alpha_{\rm \min},1)>0.$$

This conjecture is motivated by numerical computations, which also suggest that for even $k$ the minimum occurs at $\alpha_{\rm \min}=0$ and that the second derivative tends to $2$ as $k$ tends to infinity. The resolution status is not specified in the source.