Strict inequality conjecture for permutation pattern variance

Let $m>0$ and let $\tau\in S_m$. Define $L(m,\tau)$ and $R(m,\tau)$ to be the left- and right-hand sides, respectively, of equation; equivalently, the inequality is

$$\sum_{i,j=1}^{m}[i-1,j-1]_{m-1}[\tau(i)-1,\tau(j)-1]_{m-1}\geq \binom{2m-1}{m-1}^2.$$

Strict inequality conjecture. The inequality in is strict for all $m>0$. This would imply that $\operatorname{Var}(X_\tau)$ has order $2m-1$ in $n$, so the standard deviation of $X_\tau$ is one-half order smaller than its expected value. The claim is presented as a stronger form of the preceding theorem; no resolution is given in the source.