Nonexistence of paradoxical sequences above 4,614

Prove that no paradoxical Collatz sequence n, T(n), …, T^j(n), with C_j(n) = 3^{q_j(n)}/2^j < 1 and T^j(n) ≥ n, exists for the map T when the first term n exceeds 4,614.

Background

The authors computationally found exactly 593 paradoxical sequences with starting n in [7, 4614] and none with 3 ≤ n ≤ 2.8×10^19. Using heuristic arguments based on delay and maximum excursion record tables, they conjecture that no additional paradoxical sequences occur beyond n = 4614.

They note this conjecture is stronger than both the Collatz conjecture and Terras’ CST conjecture; by earlier results in the paper, finiteness (or nonexistence beyond small n) of paradoxical sequences would imply these longstanding conjectures.