The generalized ballot property for labeled chips on k-ary trees

Let $N_{k,\ell}$ denote the number of labeled chips starting at the root of a $k$-ary tree in the setting of the paper, and let a stable configuration be one obtained from these chips by chip-firing. For a vertex $v$, let its children be ordered from left to right. Generalized ballot property. For every stable configuration resulting from $N_{k,\ell}$ labeled chips starting at the root, every vertex $v$, every $i\in\{1,2,\dots,N_{k,\ell-1}\}$, and every $a,b\in\{1,2,\dots,k\}$ with $a<b$, the $i$th smallest chip in the subtree rooted at the $a$th leftmost child of $v$ is less than the $i$th smallest chip in the subtree rooted at the $b$th leftmost child of $v$. This generalizes the binary ballot property from comparisons between two sides of a vertex to comparisons among all ordered child subtrees; its validity is presented as an open question.