Hyponormality conjecture for the generalized Cesàro matrix of order two

Let $M\in\mathcal{B}(\ell^2)$ be the generalized Cesàro matrix of order two with parameter $\alpha$. Hyponormality conjecture. The operator $M$ is also hyponormal for $\alpha\in(0,1)$. The preceding finite-section calculations establish positivity for the tested sections when $\alpha\geq 0$, motivating this conjecture for the remaining parameter range; its general validity is not established in the supplied text.