Real and disjoint Casimir spectrum conjecture for parabolic positive representations

Let $\mathcal{P}_\lambda^J$ be an arbitrary parabolic positive representation of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$, and let its Casimir operators have spectrum in the ambient parameter space $\mathbb{R}^n$. Casimir spectrum conjecture. The spectra of the Casimir operators of arbitrary parabolic positive representations $\mathcal{P}_\lambda^J$ are all real-valued and disjoint in $\mathbb{R}^n$. The claim is proposed after the analysis of minimal positive representations, where the Casimir spectra were found to lie outside those of the maximal case; the general parabolic case remains to be established.