The large-number falling-time conjecture

Let $\operatorname{ft}(n)$ denote the falling time associated with the accelerated $3x+1$ map. Large-number falling-time conjecture.

$$\operatorname{ft}(n)\leq 4$$

for all $n\geq 2^{500}$.

This is a strong quantitative form of the Collatz conjecture, motivated by computations and heuristics suggesting that large integers rarely have falling time at least $5$. The threshold is supported by extensive finite searches, but the universal assertion remains open.