Equivalence conjecture for virtually unipotent curves in closed aspherical 3-manifold groups

Let $M$ be a closed aspherical $3$-manifold and let $\Gamma=\pi_1(M)$. The paper's statements numbered through concern properties of $M$ and $\Gamma$ introduced earlier in the paper. Equivalence conjecture. Statements

$$through$$

are equivalent for any closed aspherical $3$-manifold $M$ and $\Gamma=\pi_1(M)$. The equivalence is known when $M$ is a closed aspherical geometric $3$-manifold, but remains conjectural for arbitrary closed aspherical $3$-manifolds.