Conjecture on independence of optimal states from the dual Rényi indices

Let $U$ be any unitary operator linking two observables, and let $\alpha,\beta>\frac12$ satisfy the duality relation

$$\text{the duality relation specified in the source}.$$

A state is called optimal for a pair $(\alpha,\beta)$ if it attains the optimal entropic uncertainty bound for those indices.

Independence conjecture. If $\rho$ is optimal for $U$ and one dual pair $(\alpha,\beta)$, then $\rho$ is also optimal for every other dual pair.

The claim excludes the extremal pair $\{\alpha,\beta\}=\{1/2,\infty\}$. The source presents numerical evidence that differently shaped optimal bounds can nevertheless be traced out by the same states.