Four-partite network decomposition conjecture

Let $\Sigma$ be a network with four boundary vertices $A_1,\ldots,A_4$, and let $I(A_i:A_j)$ denote the mutual information between $A_i$ and $A_j$ on $\Sigma$. Let $I_3=I_3(A_1:A_2:A_3)$ be the tripartite information calculated on $\Sigma$. Four-partite network decomposition conjecture. An arbitrary network $\Sigma$ with four boundary vertices $A_1,\ldots,A_4$ decomposes into six pairwise subnetworks $\Sigma_{ij}$, $1\leq i<j\leq4$, together with a remainder subnetwork $\Sigma_r$, such that each $\Sigma_{ij}$ connects only $A_i$ and $A_j$ and satisfies

$$S_{\Sigma_{ij}}(A_i)=S_{\Sigma_{ij}}(A_j)=\frac{1}{2}I(A_i:A_j),\qquad S_{\Sigma_{ij}}(A_k)=0\quad(k\neq i,j).$$

Moreover, $\Sigma_r$ is a four-partite perfect tensor network with the same tripartite information as $\Sigma$; all pairwise mutual informations vanish and

$$S_{\Sigma_r}(A_i)=\frac{1}{2}S_{\Sigma_r}(A_iA_j)=\frac{-I_3}{2}.$$

For every subsystem $s\subset\{A_i\}$, the subsystem entropies of the seven subnetworks add up to those of $\Sigma$:

$$\sum_{i<j}S_{\Sigma_{ij}}(s)+S_{\Sigma_r}(s)=S_\Sigma(s).$$

The conjecture has numerical support from direct computations for some network examples, but the theorem cited in the source is not sufficient to prove it.