The AWGN multiway relay coding-strategy dominance conjecture

Consider an additive white Gaussian noise multiway relay channel (AWGN MWRC) with relay power $P_0=LP$, where $L$ is the number of users and $P$ is the user power. Let $R_\text{CF}$, $R_\text{CDF}$, and $R_\text{FDF}$ denote the achievable rates of compress-forward, complete-decode-forward, and functional-decode-forward, respectively. Coding-strategy dominance conjecture. For any $L$ and any $P$,

$$R_\text{CF} < \max \{ R_\text{CDF}, R_\text{FDF} \}.$$

The conjecture asserts that compress-forward is never the best of these three strategies for the equal-rate AWGN multiway relay channel under the specified power relation. The paper's numerical observations support the claim across different numbers of users and SNR values, but no general proof is supplied.