What is the latest verified research on the 3x+1 Problem?

Question

Wikipedia : Collatz Conjecture

Take any positive integer n. If n is even, divide it by $2$ to get $n / 2$. If n is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach $1$.

To quote:

Steiner (1977) proved that there is no 1-cycle other than the trivial (1;2). Simons (2004) used Steiner's method to prove that there is no 2-cycle. Simons & de Weger (2003) extended this proof up to 68-cycles: there is no k-cycle up to k = 68. Beyond 68, this method gives upper bounds for the elements in such a cycle: for example, if there is a 75-cycle, then at least one element of the cycle is less than $2385\times 2^{50}$.

Has there been any legitimate progress since then, in terms of cycles or anything else?

Answer 1

Fields Medalist Tao has a blog post on The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 from August 25, 2011.

Quoting Tao:

"Needless to say, I didn’t solve the problem, but I have a better appreciation of why the conjecture is (a) plausible, and (b) unlikely be proven by current technology, and I thought I would share what I had found out here on this blog."