  We investigate the baryon content of the circumgalactic medium (CGM) within
the virial radius of $M_h \sim 10^{12} \; M_\odot$ haloes at z ~ 3, by
modelling the surface brightness profile of the giant Ly$\alpha$ nebulae
recently discovered by MUSE around bright QSOs at this redshift. We initially
assume fluorescent emission from cold photo-ionized gas confined by the
pressure of a hot halo. Acceptable CGM baryon fractions (equal or smaller than
the cosmological value) require that the cold gas occupies $\lesssim$ 1% of the
volume, but is about as massive as the hot gas. CGM baryon fractions as low as
30% of the cosmic value, as predicted by some strongly ejective feedback models
at this redshift, are not easy to reconcile with observations, under our
assumptions, unless both the QSO-hosting haloes at $z\sim3$ are more massive
than recent BOSS estimates based on clustering and the photo-ionized gas is
colder than expected in a standard QSO ionizing radiation field. We also
consider the option that the emission is dominated by photons scattered from
the QSO broad line region. In this scenario, a very stringent lower limit to
the baryon fraction can be obtained under the extreme assumption of optically
thin scattering. We infer in this case a baryon fraction of at least 70% of the
cosmic value, for fiducial parameters. Lower values require halo masses or gas
temperatures different than expected, or that some mechanism keeps the cold gas
systematically over-pressured with respect to the ambient medium.
