Finiteness conjecture for torsion-free orbit depth and orbit depth

Let $\gamma$ be a loop in a typical fiber of the polynomial map under consideration. The torsion-free orbit depth $k$ and the orbit depth $\kappa$ are the associated depths defined through the lower central sequence and its action on the relevant homology group.

Finiteness conjecture. For any $\gamma$, the torsion-free orbit depth $k$ and the orbit depth $\kappa$ are finite.

These finiteness properties would ensure that the iterated-integral bounds for the first nonzero Melnikov function are finite. The source does not provide a proof or a counterexample, so the conjecture remains open.