Hyponormality conjecture for Cesàro matrices of positive integer order

Let $N>4$ be a positive integer. The Cesàro matrix $C(N)$ has entries

$$m_{ij}=\left\{\begin{array}{lll}\dfrac{N\prod_{t=1}^{N-1}(i+t-j)}{\prod_{t=1}^{N}(i+t)} & \text{for} & 0\leq j\leq i,\\ 0 & \text{for} & j>i,\end{array}\right.$$

and define the diagonal operator

$$P=P(N):=\operatorname{diag}\left\{\dfrac{\prod_{t=1}^{N}(n+t)}{\prod_{t=N+1}^{2N}(n+t)}:n\geq0\right\}.$$

Hyponormality conjecture for Cesàro matrices. The operator $C(N)$ is bounded and posinormal on $\ell^2$, with interrupter $P$, and consequently is also hyponormal and coposinormal. The conclusion is known for $N=1,2,3,4$; the general case $N>4$ is conjectural.