The loop-erased random-walk up-to-constants passage estimate

Let $z\in\mathbb{D}$, let $\epsilon>0$ be sufficiently small, and let $Z$, $W^w$, $\xi_x$, and $\sigma_{\mathbb{D}}$ have the meanings used in the loop-erased-random-walk estimates: $Z$ is the relevant random walk, $\xi_x$ is its hitting time of $x$, $W^w$ is the walk started at $w$, and $\sigma_{\mathbb{D}}$ is the exit time from the disk. Let $\operatorname{L}(Z[0,\xi_x])$ be the loop-erasure of $Z[0,\xi_x]$, and let $\operatorname{Es}(\epsilon n)$ be the corresponding nonintersection probability. Up-to-constants passage conjecture. For the points $w$ and $x$ in the configuration under consideration,

$$\mathbb{P}\left(\operatorname{L}(Z[0,\xi_x])\cap W^w[1,\sigma_{\mathbb{D}}]=\emptyset\right)\asymp\operatorname{Es}(\epsilon n).$$

This estimate is proposed as the missing lower-bound ingredient for the conditional occupation-measure estimate. The source had established only upper bounds and suggested that the lower bound might follow from modifying the proof strategy.