Rényi information ordering conjecture for ranked set samples

Let $\alpha>1$ and let $n\in\mathbb{N}$. Write ${\rm H}_{\alpha}(\textbf{X}_{RSS})$, ${\rm H}_{\alpha}(\textbf{X}^*_{RSS})$, and ${\rm H}_{\alpha}(\textbf{X}_{SRS})$ for the Rényi information measures of perfect ranked set sampling, imperfect ranked set sampling, and simple random sampling data, respectively. Rényi information ordering conjecture. For any $\alpha>1$ and all $n\in\mathbb{N}$, one has

$${\rm H}_{\alpha}(\textbf{X}_{RSS}) \leq {\rm H}_{\alpha}(\textbf{X}^*_{RSS})\leq {\rm H}_{\alpha}(\textbf{X}_{SRS}).$$

The analogous ordering is proved in the paper for $0<\alpha<1$, while the case $\alpha>1$ remains unsolved; examples are given in support of the conjecture.