Integral vanishing of compactly supported homology classes for the Loch Ness monster surface

Determine whether, for the Loch Ness monster surface L (the unique infinite-genus surface with one end and no boundary), the inclusion Map_c(L) -> Map(L) induces the zero map on integral homology in all positive degrees; equivalently, decide whether any compactly supported homology class of Map_c(L) survives in H_*(Map(L); Z).

Background

Theorem \ref{mainthm-LochNess} establishes that, for any field K, the inclusion Map_c(L) -> Map(L) induces the zero map on homology in positive degrees, so all dual Miller–Morita–Mumford classes are sent to zero rationally. The authors point out that this vanishing with field coefficients does not automatically imply vanishing with integral coefficients.

They explicitly note that it remains unknown whether the same conclusion holds when homology is taken with integral coefficients (Z), highlighting a gap between the field-coefficient results and the integral case.