Nonexistence of large paradoxical Collatz sequences

Let $T$ be the Collatz map. A sequence $n,T(n),\ldots,T^j(n)$ is paradoxical when $C_j(n)<1$ and $T^j(n)\geq n$. Nonexistence conjecture for large paradoxical sequences. There is no paradoxical sequence for $T$ whose first term is greater than $4614$. The conjecture is motivated by heuristic extrapolation of computational bounds; the supplied text does not establish it.