Homotopically equivalent simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links (II)

In this paper and its prequel, we give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be peripheral or torsion in the orbifold. The prequel treated the case when the 2-bridge link is a $(2,p)$-torus link, and this paper treats the remaining cases.