Boundary-cell characterization conjecture for Wilson loop diagrams

Let $\mathcal{W}(k,n)$ be the cell complex associated to admissible Wilson loop diagrams with $k$ propagators on $n$ vertices. Let $B$ be a cell of dimension $3k-1$, let $W$ and $W'$ be distinct Wilson loop diagrams, and let $\Sigma(W)$ and $\Sigma(W')$ be their associated cells. For an admissible Wilson loop diagram $\widehat W$ with $k$ propagators on $n$ vertices, let $\partial_{\widehat p,\widehat v}(\widehat W)$ be its boundary diagram and let $\partial_{\widehat v,\widehat p}(\widehat W)$ denote the corresponding minor; write $I(\widehat W)(\mathcal{Z}_*)$ for the associated integral. Boundary-cell characterization conjecture. If $B\subseteq\Sigma(W)\cap\Sigma(W')$, then the following are equivalent: (1) $B$ can be realized as the cell parametrized by some boundary diagram $\partial_{\widehat p,\widehat v}(\widehat W)$; and (2) the minor $\partial_{\widehat v,\widehat p}(\widehat W)$ corresponds to a simple pole of $I(\widehat W)(\mathcal{Z}_*)$. This is supported by the complete characterization in the case $k=2$, $n=6$; the general case remains open.