Asymptotic extremal covariance conjecture for permutation patterns

For each $m$, let $L(m,\tau)$ and $R(m,\tau)$ denote the left- and right-hand sides of equation, respectively, and define

$$M^\ast(m)=\max_{\tau\in S_m}\bigl(L(m,\tau)-R(m,\tau)\bigr),$$ $$M_\ast(m)=\min_{\tau\in S_m}\bigl(L(m,\tau)-R(m,\tau)\bigr).$$

Asymptotic extremal covariance conjecture.

$$\exists\lim_{m\to\infty}\frac{M_\ast(m)}{M^\ast(m)}=0.$$

The conjecture concerns the ratio between the minimum and maximum of the variance-related expression over permutations. The source gives no resolution.