The quasi-loop theorem for Euclidean nets

Let $G$ be a Euclidean net. For an isotropic graph of dimension two or three, the paper's theorem asserts that for every $0<\epsilon<1$ there is a $\delta=\delta(\epsilon,G)>0$ such that, for all $v\in G$, all $r>C(\epsilon,G)$, and every subset $v\in\mathcal{D}\subset B(v,r)$,

$$\mathbb{E}^{v}\operatorname{QL}(r^{1-\epsilon},r^{1-\delta},\operatorname{LE}(R[0,T(\partial\mathcal{D})]))\leq C(\epsilon,G)r^{-\delta}.$$

Quasi-loop theorem conjecture. Theorem 4.6 should hold for any Euclidean net. Extending the result would give quasi-loop control for loop-erased random walks on all Euclidean nets, rather than only the isotropic graphs covered by the theorem.