I managed to reduce a tricky problem to showing the following:
Each square of an infinite board is independently colored either black or white with probability $1/2.$ For a square $X,$ let $|X|$ denote the size of the monochromatic region containing $X.$ Prove that $\mathbb{E}\left[\frac{1}{|X|}\right] \ge 1/8.$
Through careful analysis, I found that the probability the region has size $1,2,3,4$ is $1/16, 1/32, 15/2^9, 35/2^{10}$ respectively. This means the expected value is at least $\frac{1}{16} + \frac{1}{64} + \frac{5}{2^9} + \frac{35}{2^{12}} \approx 0.0964.$ Unfortunately, this is not large enough, and I have no idea about how close the infinite sum is to $1/8.$ Finding a closed form for $p(|X|=k)$ seems infeasible. Does anyone have an idea for how to proceed?