Finite and macroscopic loop decomposition conjecture

Let $(\Lambda,\mathcal E)$ be a finite periodic box, let $\beta>0$, and let $L_1(\omega),L_2(\omega),\ldots$ be the loop lengths in decreasing order. The normalized loop lengths are $L_i(\omega)/(\beta|\Lambda|)$. Finite and macroscopic loop decomposition conjecture. There exists $\nu\in[0,1]$ such that

$$\lim_{K\to\infty}\lim_{|\Lambda|\to\infty}\sum_{i:L_i(\omega)<K}\frac{L_i(\omega)}{\beta|\Lambda|}=1-\nu$$

and

$$\lim_{k\to\infty}\lim_{|\Lambda|\to\infty}\sum_{i=1}^{k}\frac{L_i(\omega)}{\beta|\Lambda|}=\nu.$$

Thus only finite and macroscopic loops contribute in the thermodynamic limit. The conjecture is known with $\nu=0$ in dimensions $d=1,2$ and at high temperature in $d\geq3$; it is expected that $\nu>0$ for sufficiently low temperature in dimensions $d\geq3$.