The local limit conjecture for the Mahonian distribution on words

Let $N=e_1=a_1+\dots+a_m$ be the word length, let $a^*=\max_j a_j$, and set $N_*=N-a^*$. For a uniformly random word with $a_j$ occurrences of the letter $j$, let $M_{a_1,\dots,a_m}$ be its number of inversions, with mean $\mu=\mu(a_1,\dots,a_m)$ and standard deviation $\sigma=\sigma(a_1,\dots,a_m)$.

Local limit conjecture. Uniformly for all choices of $a_1,\dots,a_m$ and all integers $k$,

$$\mathcal P(M_{a_1,\dots,a_m}=k)=\frac{1}{\sqrt{2\pi}\sigma}\left(\exp\left(-\frac{(k-\mu)^2}{2\sigma^2}\right)+O\left(\frac{1}{N_*}\right)\right).$$

This would refine the paper's asymptotic normality theorem to a local limit theorem for the Mahonian inversion distribution; the conjecture remains open in the source.