Monotonicity conjecture for Rényi generalizations of conditional quantum mutual information

Let $\rho_{ABC}\in\mathcal{S}(\mathcal{H}_{ABC})_{++}$, $\tau_{AC}\in\mathcal{S}(\mathcal{H}_{AC})_{++}$, $\theta_{BC}\in\mathcal{S}(\mathcal{H}_{BC})_{++}$, and $\omega_C\in\mathcal{S}(\mathcal{H}_C)_{++}$. Let $\Delta_\alpha$ and $\widetilde{\Delta}_\alpha$ denote the Rényi core quantities derived from the families of quantities defined in the source. Monotonicity conjecture. All such Rényi core quantities are monotone non-decreasing in $\alpha$: for $0\leq\alpha\leq\beta$,

$$\Delta_{\alpha}\left(\rho_{ABC},\tau_{AC},\omega_C,\theta_{BC}\right)\leq\Delta_{\beta}\left(\rho_{ABC},\tau_{AC},\omega_C,\theta_{BC}\right),$$ $$\widetilde{\Delta}_{\alpha}\left(\rho_{ABC},\tau_{AC},\omega_C,\theta_{BC}\right)\leq\widetilde{\Delta}_{\beta}\left(\rho_{ABC},\tau_{AC},\omega_C,\theta_{BC}\right),$$

and analogous inequalities hold for all orderings of the last three arguments of $\Delta_\alpha$ and $\widetilde{\Delta}_\alpha$. The conjecture concerns the Rényi generalizations of conditional quantum mutual information. It was proved for $\alpha$ in a neighborhood of one and in some other special cases, but remains open in general.