Multi-trek separation conjecture for higher-order moment tensors

Let $G$ be a directed acyclic graph with node set $V$, and let $k>3$. For subsets $S_1,\ldots,S_k\subseteq V$ with

$$\#S_1=\cdots=\#S_k,$$

write $\mathcal{N}^{(k)}_{S_1,\ldots,S_k}$ for the subtensor of the $k$th-order moment tensor indexed by these sets. A $k$-split-trek system between $S_1,\ldots,S_k$ is a system of $k$-split-treks, and it has a sided intersection when two of its constituent paths intersect on the same side. Multi-trek separation conjecture. The determinant satisfies

$$\det\mathcal{N}^{(k)}_{S_1,\ldots,S_k}=0$$

if and only if every system of $k$-split-treks between $S_1,\ldots,S_k$ has a sided intersection. The analogous criterion is proved for third-order moments, but for $k>3$ the relevant tensor is not diagonal, so the extension to higher-order moments remains conjectural.