Capacity characterization for group leaders satisfying the receiver side-information condition

Consider a private-message AWGN broadcast channel with $Q>3$ receivers, ordered by noise levels as $N_1\leq N_2\leq\cdots\leq N_Q$. A group leader is the common subgraph obtained from a group of side-information graphs by removing all arcs from a stronger receiver to a weaker receiver. For receiver $i$, let $\mathbf{K}_i$ denote the set of messages known to receiver $i$. Capacity characterization conjecture. If, for every receiver $i$ in a group leader, each side-information message $M_j\in\mathbf{K}_i$ is also known to every receiver that is stronger than receiver $i$ and weaker than receiver $j$, namely every receiver $q$ satisfying $j<q<i$, then the capacity region can be established for all members of the group. The capacity region is achieved using Codebook Construction A and Decoding Scheme A.