Oh, right … I meant to say 5n+1 instead of 3n+1.
For 3n+1, a 50/50 split of 0s and 1s will crash any start number to 1. (Assuming \pi is normal, which, nobody knows.)
Speaking of 5n+1, @Failix posted the idea that a divergent 5n+1 trajectory’s parity sequence has to have at least some minimal amount of 0’s. I wonder if a 50/50 split is okay (e.g., doesn’t involve too much “loitering”).