Uniqueness conjecture for the minimizing parameter

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 $\lambda_0(\alpha,1)$. A minimizing parameter is an $\alpha_{\rm \min}\in\mathbb{R}$ satisfying $\lambda_0(\alpha_{\rm \min},1)=\hat{\nu}$. Uniqueness conjecture. There exists a unique $\alpha_{\rm \min}\in\mathbb{R}$ such that

$$\lambda_0(\alpha_{\rm \min},1)=\hat{\nu}.$$

The conjecture is motivated by numerical computations. The source notes that uniqueness had been claimed for $k=1$ by Pan–Kwek and for arbitrary odd $k$ by Aramaki, but does not establish the claim or provide a resolution status.