Planar basis evaluation conjecture for $\eta_J$ on minimal kinematics

Let $\Delta_{k,n}$ be the $k$th hypersimplex. For a $k$-element subset $J\subset\{1,\dots,n\}$, let $\Pi_J$ be the associated polytope obtained by eliminating the block containing $n$, and let $I_a$ and $I_b$ be the subsets constructed from the cyclic block decomposition of $J$, with $I_b=J\setminus S_{j-1}$ and $I_a$ the interval of length $k-s_{j-1}$ starting at the cyclically smallest element of $S_j$. Let $\eta_J$ be the corresponding planar basis element, let $x_i$ denote the minimal-kinematics variables, and call $J$ frozen when $\eta_J$ vanishes on minimal kinematics. Planar basis evaluation conjecture. For every nonfrozen $k$-element subset $J$ of $\{1,\dots,n\}$,

$$\left.\eta_J\right|_{\mathrm{MK}}=\sum_{i\in I_a}x_i-\sum_{i\in I_b}x_i.$$

Moreover, $I_a\setminus I_b$ and $I_b\setminus I_a$ are non-crossing; specifically, $i<j$ for every $i\in I_a\setminus I_b$ and $j\in I_b\setminus I_a$. For frozen $J$, one has $\left.\eta_J\right|_{\mathrm{MK}}=0$. The conjecture gives the proposed evaluation of planar basis elements on minimal kinematics; the supplied text does not state a resolution, so its status remains open.