The ballot-property conjecture for stable configurations on undirected binary trees

Let a stable configuration be a chip distribution on an undirected binary tree, and say that it satisfies the ballot property if, for each $i\in\mathbb{N}$, the $i$th smallest chip in the subtree rooted at the left child of a subtree's root has a smaller label than the $i$th smallest chip in the subtree rooted at the right child. A subtree means any subtree of the tree.

Ballot-property conjecture. In a stable configuration, the whole tree and every subtree satisfy the ballot property.

The paper establishes the property for subtrees with two or three layers and reports computational verification for every stable configuration starting with $15$ chips. The general assertion remains unresolved in the supplied text.