Conjecture on the levels and adjacency of infinite connected components

Consider the infinite connected components of the corner-percolation configuration, each assigned its integer-valued level. Level-bijection conjecture. The function which maps each infinite connected component to its level is bijective. Moreover, two infinite connected components are neighbors if and only if the difference of their levels is $-1$ or $1$.

This conjecture describes the global ordering of infinite connected components by their levels and predicts that adjacency occurs exactly between consecutive levels. The supplied text gives no resolution of the conjecture.