Higher-order cluster-expansion conjecture for the Ising model on even tori

Fix $k\in\mathbb{N}$. For the Ising partition function $Z_{\mathbb{Z}_m^t}(\lambda,\beta)$ on the even torus $\mathbb{Z}_m^t$, let $L_j$ denote the sum of the weights of clusters of size $j$ on the even side,

$$L_j=\sum_{\Gamma\in\mathcal{C}_{\mathcal{E},j}}\omega(\Gamma).$$

Higher-order cluster-expansion conjecture. There exist functions $f_k(t)$ and $g_k(t)$ with $f_k(t)=\widetilde{\Omega}(t^{-1/2})$ and $g_k(t)=o_t(1)$ such that, for every $\lambda>0$, $\beta\in(0,\infty]$ satisfying $\lambda(1-e^{-\beta})\geq f_k(t)$, and every even $m$,

$$Z_{\mathbb{Z}_m^t}(\lambda,\beta)=2(1+\lambda)^{m^t/2}\exp\left(\frac{m^t}{2}\lambda\left(\frac{1+\lambda e^{-\beta}}{1+\lambda}\right)^{2t}+L_2+\dots+L_k(1+g_k(t))\right).$$

The conjecture extends the range in which finitely many cluster-expansion terms give the exponential asymptotics; the paper proves analogous results for the regimes covered by its methods, but leaves this broader statement open.