The orbit-depth dichotomy for polynomial Hamiltonians and Melnikov-function length

Let $F$ be a polynomial, let $\gamma$ be a non-trivial cycle of $F$, and let the orbit depth be the depth defined from the monodromy orbit of $\gamma$ and the lower central sequence of the fundamental group. For a deformation $\omega$, let $M_\mu$ denote its first non-zero Melnikov function and let its length be the iterated-integral length.

Orbit-depth and Melnikov-length conjectures. (i) For any polynomial $F$ and any non-trivial cycle $\gamma$ of $F$, either the depth is unbounded, or it is $1$, or $2$. (ii) For any $F$ and its cycle $\gamma$, either there exist deformations $\omega$ whose first non-zero Melnikov function $M_\mu$ has arbitrarily high length, or, for any deformation $\omega$, the length of the first non-zero Melnikov function is $1$ or $2$.

These conjectures propose a sharp dichotomy between unbounded orbit depth and the two smallest bounded depths, and relate the possible depth directly to the complexity of Melnikov functions. The paper gives examples with unbounded depth and establishes the relevant bounds in several configurations of products of four real lines, but the general assertions remain open.