Generalize the residueless-stratum argument to the remaining genus-three signatures

Develop a generalization of the arguments used to analyze the genus-one residueless strata R(μ,-2,-2) for the signatures μ=(4), (2,2), and (3,1) to the remaining signatures μ=(2,1,1) and μ=(1,1,1,1).

Background

The paper proves the Chow–Künneth generation property for the multi-scale compactifications of the genus-three holomorphic strata with signatures (4), (2,2), and (3,1). A key technical ingredient is the analysis of the genus-one residueless strata R(μ,-2,-2), which arise in the boundary of these compactifications. For the three treated signatures, the arguments are ad hoc and rely on the fact that these residueless strata have dimension at most two.

The remark identifies the unresolved extension of this method to the signatures μ=(2,1,1) and μ=(1,1,1,1). Although the authors note that suitable S₂-quotients of the corresponding compactifications admit birational morphisms to \overline{M}{3,1} and \overline{M}{3,2}, respectively, the stated open issue is how to generalize the specific residueless-stratum arguments themselves to these remaining signatures.