Golden Automaton conjecture on inductive Collatz convergence

Let $n$ be a positive integer, and consider the Collatz dynamic on odd positive integers. The Golden Automaton is the finite proof system described in the source for establishing Collatz convergence. Golden Automaton conjecture. If all odd numbers up to $2^n$ are proven to converge to $1$ under the Collatz dynamic, then the Golden Automaton finitely proves the convergence of all odd numbers up to $2^{n+1}$. This would provide an inductive mechanism for extending verified Collatz convergence to larger ranges, but the source does not state that the conjecture has been proved or disproved.