Scaling conjecture for modified cluster-expansion limits

Let $m=\alpha n$ with fixed proportionality parameter $\alpha$, and let $\hat T_i(n)$ be the modified cluster-expansion terms. Let $Q_i(r)$ denote the limits from the unmodified expansion.

Modified scaling conjecture. The limits obtained using $E_1$, $E_2$, or $E$ are equal and satisfy

$$\lim_{n\to\infty}\frac{1}{n}\hat T_i(n)\equiv\hat Q_i(r)=\alpha Q_i(r/\alpha).$$

This conjecture specifies the dependence of the modified limits on $\alpha$ by rescaling the parameter in the earlier limits. It is proposed without proof in the paper.