At that point, the conjecture's just numerology: 27 takes 5 bits, and 9232 takes 14 bits (two shy of 3×5+1 = 16). 27 is the peak of the average ratio between start and maximum, because the +1s are so significant when the numbers are small: past that point, we're relying on extreme outlier behaviour to get each new high-score. Those only start showing up often enough to matter once we get into the thousands.

Plugging in values from OEIS A006884, it looks like the maximum ratio between the maximum and starting values goes down until around 4255, then picks up again, gradually increasing from there. Eyeballing the growth rate, I suspect there's a counterexample to this interpretation somewhere before 10^1000. (Does anyone have an element of A006884 greater than 2358909599867980429759? That's 140 bits maximum to 71 bits starting.)