One potential counterexample of the Collatz conjecture would be if there was a number that looped back to itself. Of course, this would still not prove the conjecture because some sequences could still potentially diverge to infinity. Has it been proven that Collatz sequences cannot repeat?
No but it has been proven that if a cycle exist it is large and has certain properties.
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 2385×2^50... from collatz wiki
from the connection between cycle length and maximum number in cycle they can be exaustivly searched and cycle of that length disproven. but no general result.