Logarithmic and polynomial bounds for delay and maximum excursion

Establish the existence of absolute constants α, β > 0 such that, for every integer n ≥ 2 under the Collatz map T, the delay satisfies d_T(n) ≤ α log n and the maximum excursion satisfies M_T(n) ≤ n^β.

Background

The delay d_T(n) is the least j with T^j(n) = 1, and the maximum excursion M_T(n) is the maximum value attained along the Collatz orbit. The authors refer to heuristic models and empirical data (e.g., work by Kontorovich and Lagarias, and computational records) suggesting such bounds should hold, at least asymptotically.

These bounds, if proved, would imply finiteness of paradoxical sequences above 2 via inequalities developed in the paper that relate ones-ratios, harmonic means of odd terms, and iteration counts.