The intersection-number-zero criterion for plabic graphs

Fix a positive integer $m$, and let $G$ be a plabic graph of type $(k,n)$ with $\dim\Pi_G=km$. Let $T_G$ denote the associated family of $k$-planes. Intersection-number-zero criterion conjecture. The following are equivalent: (i) $\Pi_G$ has $m$-intersection number $0$; (ii) there is a plabic graph $G'$ of type $(k',n')$ with $1\leq k'<k$ and $\dim\Pi_{G'}<k'm$ such that every $V\in T_G$ contains some $V'\in\Pi_{G'}$; (iii) a graph move-equivalent to $G$ has a subgraph $G'$ and perfect orientation satisfying the assumptions of the sufficient criterion in Proposition 1. The equivalence is conjectural, based on computations for $m\leq 8$.