Parallel-curve characterizations of orbit-group solvability and orbit-depth boundedness

Let $F$ be a Hamiltonian whose foliation contains curves that may be parallel, let $\pi_1$ be the fundamental group of a regular fiber, and let $\mathcal{O}$ be the monodromy orbit subgroup generated by a cycle. Write $\pi_1^{\mathcal{O}\text{-ab}}$ for the corresponding orbit-abelian quotient. The orbit depth is the depth defined using $\mathcal{O}$ and the lower central sequence of $\pi_1$.

Parallel-curve characterization conjecture. (i) The non-solvability of the group $\pi_1^{\mathcal{O}\text{-ab}}$ is characterized by the presence of a pair of parallel curves in the Hamiltonian foliation. (ii) The non-bounded orbit depth is characterized by the presence of two pairs of parallel curves in the Hamiltonian foliation. (iii) The type of singularity given by a pair of parallel curves at the line at infinity characterizes the non-solvability of $\pi_1^{\mathcal{O}\text{-ab}}$.

The claims seek geometric criteria for algebraic properties of the orbit-abelian quotient and for boundedness of orbit depth. The paper proves results for products of four real lines, including the trapezoid and parallelogram cases, but the stated characterizations are not established in full generality.