  We present the results from a MUSE survey of twelve $z\simeq3.15$ quasars,
which were selected to be much fainter (20<i<23) than in previous studies of
Giant Ly$\alpha$ Nebulae around the brightest quasars (16.6<i<18.7). We detect
HI Ly$\alpha$ nebulae around 100% of our target quasars, with emission
extending to scales of at least 60 physical kpc, and up to 190 pkpc. We explore
correlations between properties of the nebulae and their host quasars, with the
goal of connecting variations in the properties of the illuminating QSO to the
response in nebular emission. We show that the surface brightness profiles of
the nebulae are similar to those of nebulae around bright quasars, but with a
lower normalization. Our targeted quasars are on average 3.7 magnitudes (~30
times) fainter in UV continuum than our bright reference sample, and yet the
nebulae around them are only 4.3 times fainter in mean Ly$\alpha$ surface
brightness, measured between 20 and 50 pkpc. We find significant correlations
between the surface brightness of the nebula and the luminosity of the quasar
in both UV continuum and Ly$\alpha$. The latter can be interpreted as evidence
for a substantial contribution from unresolved inner parts of the nebulae to
the narrow components seen in the Ly$\alpha$ lines of some of our faint
quasars, possibly from the inner CGM or from the host galaxy's ISM.
