The capacity trichotomy conjecture for three-source three-terminal sum-networks

Let a sum-network have $m=n=3$, where $m$ is the number of sources and $n$ is the number of terminals, and let its capacity be the supremum of the achievable communication rates for computing the sum of the source messages at every terminal.

Capacity trichotomy conjecture. The capacity of the sum-network is either $0$, $2/3$, or at least $1$.

The preceding discussion gives examples and structural observations supporting this classification, but the statement is presented as a belief rather than established here. No resolution is supplied in the source.