Pairwise structure conjecture for equal smallest minors

Let $I,J\in\binom{[n]}{k}$, let $r=|I\setminus J|=|J\setminus I|$, and form the lattice path $P(I,J)$ from the ordered symmetric difference, with an up step for an element of $I\setminus J$ and a down step for an element of $J\setminus I$. Assume $P(I,J)$ is a Dyck path. A pair is $p$-interlaced when the path has $p$ peaks, and in the 2-interlaced case write its successive up- and down-run lengths as $(\alpha_1,\beta_1,\alpha_2,\beta_2)$.

Pairwise smallest-minors conjecture. There exists an arrangement of smallest minors $S\subset\binom{[n]}{k}$ containing $I$ and $J$ if and only if either the pair is 1-interlaced, equivalently weakly separated, or it is 2-interlaced and

$$\alpha_i\ne\beta_j$$

for every $i$ and $j$.

This conjecture describes which pairs can occur together among equal minimal Plücker coordinates. The paper studies the conjecture through plabic graphs and proves related cases and constructions, but the supplied text gives no resolution of this exact statement.