Critical polynomial decay conjecture for the loop O(1) model on higher-dimensional lattices

Let $d\geq 3$, let $x_c$ be the critical loop-model parameter on $\mathbb{Z}^d$, and let $\mathcal{C}_0$ be the cluster containing the origin. Write $\ell_{x_c,\mathbb{Z}^d}$ for the corresponding loop $\mathrm{O}(1)$ measure.

Critical polynomial decay conjecture. One has

$$\ell_{x_c,\mathbb{Z}^d}[|\mathcal{C}_0|]<\infty,$$

and there exist $a,b>0$ such that

$$\frac{a}{|v|^b}<\ell_{x_c,\mathbb{Z}^d}[0\leftrightarrow v].$$

At criticality, the paper expects finite expected cluster size together with polynomially decaying, rather than exponentially decaying, connection probabilities. Polynomial lower bounds are accessible on the torus and in dimension two, but transferring them to $\mathbb{Z}^d$ for $d\geq 3$ remains open.