Asymptotic conjecture for Coxeter sorting time

Let ${\operatorname{Sort}}_n$ denote the Coxeter sorting time for permutations of size $n$. Asymptotic sorting-time conjecture.

$${\operatorname{Sort}}_n=\frac{8-\pi}{16}\cdot n^2 +O(n).$$

This conjecture seeks an asymptotically sharp estimate for the sorting time; the source presents it as an improvement over the preceding theorem, and no resolution is supplied.