Capacity region of the linear deterministic broadcast channel with a diamond message set

Let $K$ be the number of receivers, let $\mathsf{E}=\{M_{\overline{\phi}},M_{\overline{K}},M_{\overline{K-1}},M_{\overline{K-1.K}}\}$ be the diamond message set, and let $r_S$ denote the rank associated with a receiver-index set $S=l_1l_2\cdots l_N\subseteq [1:K]$:

$$r_S=\operatorname{rank}\left(\begin{bmatrix}\boldsymbol{H}_{l_1}\\ \vdots \\ \boldsymbol{H}_{l_N}\end{bmatrix}\right).$$

The message rates are $R_{\overline{\phi}},R_{\overline{K}},R_{\overline{K-1}},R_{\overline{K-1.K}}$.

The capacity-region claim. The capacity region of the $K$-user linear deterministic broadcast channel for this diamond message set is the set of non-negative rate tuples satisfying, for every $j\in\{1,2,\cdots,K-2\}$,

$$R_{\overline{\phi}}+R_{\overline{K-1}}\leq r_K,$$ $$R_{\overline{\phi}}+R_{\overline{K}}\leq r_{K-1},$$ $$R_{\overline{\phi}}+R_{\overline{K-1}}+R_{\overline{K}}\leq r_{K-1.K},$$ $$R_{\overline{\phi}}+R_{\overline{K-1}}+R_{\overline{K}}+R_{\overline{K-1.K}}\leq r_j,$$ $$2R_{\overline{\phi}}+R_{\overline{K-1}}+R_{\overline{K}}+R_{\overline{K-1.K}}\leq r_{j.K-1.K}-r_{K-1.K}+r_{K-1}+r_K,$$ $$2R_{\overline{\phi}}+2R_{\overline{K-1}}+2R_{\overline{K}}+R_{\overline{K-1.K}}\leq r_{j.K-1}+r_{j.K}+r_{K-1.K}-r_{j.K-1.K}.$$

The claim gives an exact characterization of the achievable rates for this deterministic broadcast-channel message configuration; the supplied text does not indicate whether the result is intended as a conjecture or has an independently established resolution.