  We show that a conjecture of Putman--Wieland, which posits the nonexistence
of finite orbits for higher Prym representations of the mapping class group, is
equivalent to the existence of surface-by-surface and surface-by-free groups
which do not virtually algebraically fiber. While the question about the
existence of such groups remains open, we will show that there exist
free-by-free and free-by-surface groups which do not algebraically fiber (hence
fail to be virtually RFRS).
