The general DoF upper-bound conjecture for cached two-way 2×2×2 relay networks

Consider a two-way $2\times2\times2$ relay network with two multiple-antenna relays having $N_1$ and $N_2$ antennas, respectively, and four single-antenna transceivers. Let ${\text{DoF}}_{C}$ denote the total degrees of freedom when the relays use caching.

General DoF upper-bound conjecture. The total degrees of freedom satisfies

$${\text{DoF}}_{C}\le \frac{4(N_1+N_2)}{N_1+N_2+1}.$$

The preceding proposition establishes this bound when the relays transmit only the cached combined messages rather than the original individual messages. The conjecture asserts that the same expression is a general upper bound for the cached two-way relay network, including schemes using the original individual messages.