Edit: Thanks to all for your comments. Because of the phrasing of the review of Eliahou's paper on MathSciNet, I was confused between the $3n+1$ and the $3n-1$ problems. But Eliahou's paper is indeed about the $3n+1$, hence is relevant to the OP. I edited my answer so that it should now make sense.

It is known that any non-trivial cycle must start after $2^{40}$, and have length $17 087 915b + kc$, where $b,c\in\mathbb{N}$ and $k\in\{301 994 , 85 137 581\}$: see Shalom Eliahou, {\it The $3x + 1$ problem: new lower bounds on nontrivial cycle lengths}. Discrete Math. 118 (1993), 45–56.