Base-set Laurent phenomenon conjecture for minimal circular planar networks

Let $L$ be the response matrix of a minimal circular planar network with $m$ edges, and call a minor of $L$ noninterlaced when its row and column index sets are noninterlaced. Base-set Laurent phenomenon conjecture. There is a set $S$ of $m$ noninterlaced minors of $L$, analogous to the small central minors for a well-connected network, such that every nonzero noninterlaced minor of $L$ is both a Laurent polynomial in the minors from $S$ and a positive function of the minors from $S$. This would extend the Laurent-positivity description known for well-connected networks to minimal networks that are not necessarily well-connected; the source gives no resolution.