Mean-dominates-standard-deviation conjecture for the FK coupling time

Let $T_L$ be the coupling time of the FK heat-bath process on $\mathbb{Z}_L^d$, with $d\geq2$, $q\geq1$, and $p\in(0,1)$. Let $q_*$ be the threshold for the order of the phase transition, and assume additionally that $p\neq p_{\mathrm{c}}$ whenever $q\geq q_*$. Write $\mu_T(L)=\mathbb{E}(T_L)$ and $\sigma_T(L)=\sqrt{\operatorname{var}(T_L)}$.

Mean-dominates-standard-deviation conjecture.

$$\frac{\sigma_T(L)}{\mu_T(L)}\to0\qquad\text{as }L\to\infty.$$

The claim predicts concentration of the coupling time on its mean scale, including critical points with continuous transitions. Numerical evidence is given, while the paper reports that it fails at criticality for $q\geq q_*$.