The BSC-PSC bound for sandwiched Rényi conditional entropy

Let $X_1,X_2$ be binary random variables with quantum side information $B_1,B_2$, let $\rho_1,\rho_2$ denote their respective classical–quantum states, and let $X_1+X_2$ be their sum modulo two in the product state $\tau$. Write

$$H_1=\widetilde H_\alpha^\downarrow(X_1|B_1)_{\rho_1},\qquad H_2=\widetilde H_\alpha^\downarrow(X_2|B_2)_{\rho_2}.$$

Here $h_\alpha$ is the binary Rényi entropy and $h_\alpha^{-1}$ its inverse on the relevant interval, while $\ast$ denotes binary convolution. The BSC-PSC bound. For $\alpha\in(0,2]\cup[3,\infty)$,

$$\widetilde H_\alpha^\downarrow(X_1+X_2|B_1B_2)_\tau\geq\begin{cases} h_\alpha\bigl(h_\alpha^{-1}(H_1)\ast h_\alpha^{-1}(H_2)\bigr),&H_1+H_2\leq\log 2,\\ H_1+H_2-\log 2+h_\alpha\bigl(h_\alpha^{-1}(\log 2-H_1)\ast h_\alpha^{-1}(\log 2-H_2)\bigr),&H_1+H_2\geq\log 2.\end{cases}$$

For $\alpha\in[2,3]$, the same statement holds with $\geq$ replaced by $\leq$; for $\alpha\in\{2,3\}$, equality holds. If true, the bounds are tight, with equality achieved by binary symmetric and pure-state channels, but their validity for general $\alpha$ remains unresolved.