The cycle conjecture for 3-free Fibonacci sequences

A 3-free Fibonacci sequence is a Fibonacci-type sequence in which each newly appended term is obtained by dividing the sum of the preceding two terms by a power of $3$ whenever the resulting term is integral, subject to the paper's definition. Cycle conjecture. Any 3-free Fibonacci sequence ends in a cycle. The preceding probabilistic analysis suggests that such sequences tend to decrease, but the conjecture asserts eventual periodicity rather than termination.