  Recent chain-like structured materials have shown a robust superconducting
phase. These materials exhibit the presence of quasi-one-dimensional bands
(q1D) coupled to conventional higher-dimensional bands. On the mean-field level
such systems have a high critical temperature when the chemical potential is
close to the edge of a q1D band and the related Lifshitz transition is
approached. However, the impact of the pair fluctuations compromises the
mean-field results. Recently it has been demonstrated that these fluctuations
can be suppressed (screened) by a specific multiband mechanism based on the
pair-exchange coupling of the q1D condensate to a stable higher-dimensional
one. In the present work we demonstrate that strikingly enough, this mechanism
is not very sensitive to the basic parameters of the stable condensate such as
its strength and dimensionality. For example, even the presence of a passive
higher-dimensional band, which does not exhibit any superconducting
correlations when taken as a separate superconductor, results in suppression of
the pair fluctuations.
