The most likely terminal configuration conjecture for labeled chip-firing on binary trees

Consider the labeled chip-firing process on a complete binary tree with $2^n-1$ nodes, using the labels $1,2,\dots,2^n-1$, and let a terminal configuration be a configuration obtained when no further firing moves are possible. A binary search tree is a binary tree in which every node's chip is larger than all chips in its left subtree and smaller than all chips in its right subtree; for these labels, there is a unique binary search tree.

Most likely terminal configuration conjecture. For all $n$, the most likely terminal configuration is the unique binary search tree on the complete binary tree with $2^n-1$ nodes.

The conjecture is motivated by computations for $n=3$ and $n=4$, where the most likely terminal configuration was observed to be the binary search tree. The paper reports that reliable frequency data were unavailable for $n\geq5$ because the number of possible terminal configurations grows rapidly.