Small-time percolation conjecture for perturbed hexagonal packing

Let $\Pi_0$ be the hexagonal circle-packing point process, let $\Pi_t$ be obtained by moving its nodes independently according to Brownian motion for time $t$, and let $R(\Pi_t)$ be the union of balls of radius $1/2$ centered at the nodes of $\Pi_t$. Small-time percolation conjecture. There exists $t_0>0$ such that, for all $t<t_0$, $R(\Pi_t)$ contains an infinite component almost surely. This predicts that sufficiently small Brownian perturbations of the critical hexagonal packing retain percolation; the source presents it as an open problem.