Lower bounds for the total stopping time of 3x + 1 iterates

The total stopping time σ∞(n) of a positive integer n is the minimal number of iterates of the 3x + 1 function needed to reach the value 1, and is +∞ if no iterate of n reaches 1. It is shown that there are infinitely many positive integers n having a finite total stopping time σ∞(n) such that σ∞(n) γBP ≈ 41.677647, a search of all 3x + 1 trees to sufficient depth could produce a proof that there are infinitely many n such that σ∞(n) < γ log n. It would require a very large computation to search 3x + 1 trees to a sufficient depth to produce a proof that the expected behavior of a "random" 3x + 1 iterate, which is γ = 2/log4/3 ≈ 6.95212, occurs infinitely often.