Off-critical variance conjecture for the FK heat-bath coupling

Let $T_L$ be the coupling time and $W_L$ the coupon-collector time for the FK heat-bath coupling on the torus $\mathbb{Z}_L^d$, where $d\geq2$, $q\geq1$, and $p\in(0,1)$ with $p\neq p_{\mathrm{c}}$. Define $\sigma_T(L)=\sqrt{\operatorname{var}(T_L)}$ and $\sigma_W(L)=\sqrt{\operatorname{var}(W_L)}$.

Off-critical variance conjecture. There exists $C(p,q,d)\geq1$ such that, as $L\to\infty$,

$$\sigma_T(L)\sim C(p,q,d)\,\sigma_W(L).$$

The conjecture asserts coupon-collector scaling for fluctuations away from criticality; numerical evidence is presented in the paper. Under an additional relaxation-time hypothesis, it is equivalent to a relaxation-time scaling statement.