Sharp covering-decay conjecture for short paths

Let $N\ge 1$, let $\mathcal{P}$ be a nearest-neighbor path in $\mathbb{Z}^d$ connecting $0$ to $\partial B_1(0,N)$, and write $|\mathcal{P}|$ for its length. Let $\{X_n\}_{n=0}^\infty$ be a $d$-dimensional simple random walk starting at $0$.

Sharp covering-decay conjecture. There are an $\epsilon>0$ and a $\hat P_d\in(0,1)$ such that, for every $d\ge4$ and every such path satisfying $|\mathcal{P}|\le N^{1+\epsilon}$,

$$P\left({\rm Trace}(\mathcal{P})\subseteq {\rm Trace}\big(\{X_n\}_{n=0}^\infty\big)\right)\le \hat P_d^{N^{1+\epsilon}}.$$

This would improve the theorem's upper bound for paths of length $N^{1+\epsilon}$ and yield nontrivial bounds for the chemical distance within a three-dimensional random-walk trace. The conjecture is presented as open; the paper notes that its proposed reflection argument cannot prove it.