The universal crossing conjecture for square tilings

Tile the unit square with possibly infinitely many squares of varying sizes, with at most three squares meeting at each corner. Color each square black or white independently with equal probability.

Universal crossing conjecture. There is a universal constant $c>0$ such that the probability of a black left-to-right crossing is greater than $c$.

This is posed as a heuristic in the paper's discussion of conformal invariance and square tilings. The supplied text gives no resolution or further hypotheses, so the conjecture remains open.