I'm pretty sure you're sampling the "bedposts" as well, the upper and lower subgraphs aren't just connected at every node; I understood the rough (and incorrect) intuition to be like:

P[nodes in upper bunk connected] > P[nodes in lower bunk connected] * P[sufficient "bedpost" edges survive connecting upper and lower]

Since

P[nodes in upper bunk connected] = P[nodes in lower bunk connected]

And by definition

P[sufficient "bedpost" edges survive connecting upper and lower]<1