Capacity conjecture for degraded broadcast relay channels with common relay

Let a broadcast relay channel with common relay have input alphabets represented by $X,X_1,X_2$ and output alphabets $Y_1,Z_1,Y_2,Z_2$. In the degraded case, the channel satisfies the Markov chains

$$X \mathbin{-\mkern-6mu\circ\mkern-6mu-} (X_1,Z_1) \mathbin{-\mkern-6mu\circ\mkern-6mu-} (Y_1,Y_2),\qquad (X,X_1) \mathbin{-\mkern-6mu\circ\mkern-6mu-} Y_1 \mathbin{-\mkern-6mu\circ\mkern-6mu-} Y_2.$$

For auxiliary random variables $U,V$ satisfying $(V,U) \mathbin{-\mkern-6mu\circ\mkern-6mu-} (X_1,X) \mathbin{-\mkern-6mu\circ\mkern-6mu-} (Y_1,Z_1,Y_2)$, let $\mathscr{P}$ be the set of all joint distributions $P_{VUX_1X}$ with this Markov property. Capacity conjecture for the degraded broadcast relay channel with common relay. The capacity region is

$$\begin{array}{l} R_0 \leq I(U,V;Y_2),\\ R_0+R_1 \leq \min\{I(X;Z_1\mid V,X_1),I(X,X_1;Y_1)\}, \\ R_0+R_1 \leq \min\{I(X;Z_1\mid X_1,V,U),I(X,X_1;Y_1\mid U,V)\}+I(V,U;Y_2). \end{array}$$

The source states that the achievability part follows by taking $U_3=U_4=\emptyset$ and $V_0=bU_0+(1-b)X_1$, with $b$ Bernoulli of parameter $p$ from the preceding theorem; thus the displayed region is presented as a resolved capacity characterization rather than an open conjecture.