Non-crossing families and labeling sets of plabic diagrams

Let $[1\dots n]$ be labeled cyclically, and let a non-crossing family be a family of pairwise non-crossing $k$-subsets, where non-crossing means that no chord with endpoints in $I-J$ crosses a chord with endpoints in $J-I$ for two $k$-subsets $I$ and $J$. Let ${\bf D}$ be a $\pi_{k,n}$-diagram, and call the associated $k$-subsets its labeling sets. Non-crossing families conjecture. Every maximal family of non-crossing $k$-subsets is a collection of labeling sets for some $\pi_{k,n}$-diagram ${\bf D}$. In particular, every maximal family consists of $k(n-k)+1$ subsets. The preceding corollary establishes the converse inclusion for labeling sets of $\pi_{k,n}$-diagrams. The conjecture asserts that all maximal non-crossing families arise in this way; the source gives no resolution.