  Let g_t be a family of constant scalar curvature metrics on the total space
of a Riemannian submersion obtained by shrinking the fibers of an original
metric g, so that the submersion collapses as t approaches 0 (i.e., the total
space converges to the base in the Gromov-Hausdorff sense). We prove that,
under certain conditions, there are at least 3 unit volume constant scalar
curvature metrics in the conformal class [g_t] for infinitely many t's
accumulating at 0. This holds, e.g., for homogeneous metrics g_t obtained via
Cheeger deformation of homogeneous fibrations with fibers of positive scalar
curvature.
