The odd-configuration inversion conjecture

Let $n=2m+1$, and let $\widetilde{\Delta^n}$ denote the stable configurations reachable from $\Delta^n$ by chip-firing. In the odd case, these configurations may be identified with permutations, and an inversion is a pair of entries in the permutation that occur in reverse order. Odd-configuration inversion conjecture. The maximum number of inversions among all permutations in $\widetilde{\Delta^n}$ is exactly

$$m.$$

The conjecture concerns the structure of reachable stable configurations for odd $n$. It had been verified in the source for odd $n\leq9$, but no general proof was given.