I am a bit late to the party on this, but last year saw a significant advancement in our general understanding of the Collatz Conjecture. Using methods similar to those used in solving partial differential equations, Terence Tao showed that the conjecture is true for almost all numbers. This goes beyond the work done by Ivan Korec by proving it for a larger population but falls short of proving it for all numbers. Admittedly, this proof probably will not be extended to all numbers, requiring a different methodology to progress even more.
A general description of the paper was recently published by Quanta Magazine.
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