Tightness of the communication-rate upper bounds for hypergraphical sources

Let $X_{\mathcal{M}}$ be a hypergraphical source, and let $R_{\mathrm{SK}}$ denote the minimum communication rate required to achieve secret-key capacity. Theorem~ gives an upper bound on this rate for general hypergraphical sources, while Theorem~ gives the corresponding upper bound for pairwise independent network sources on weighted graphs. Tightness conjecture. The upper bounds in Theorems~ and are tight. This conjecture asserts that the displayed upper bounds exactly equal the communication rates needed to achieve secret-key capacity; the source does not provide a resolution.