Nonpercolation conjecture for Brownian-perturbed square packing

Let $\Pi_0^\mathrm{s}$ be the vertices of the square lattice with side length $1$, and let $\Pi_\epsilon^\mathrm{s}$ be obtained by moving those nodes independently according to Brownian motion for time $\epsilon$. Let $R(\Pi_\epsilon^\mathrm{s})$ be the union of balls of radius $1/2$ centered at the moved nodes. Square-packing nonpercolation conjecture. For any $\epsilon>0$, all components of $R(\Pi_\epsilon^\mathrm{s})$ are finite almost surely. This conjecture asserts that every positive Brownian perturbation destroys percolation in the square packing, motivated by the subcritical edge-intersection probability described in the source.