  Enormous Ly$\alpha$ nebulae, extending over 300-500\,kpc around quasars,
represent the pinnacle of galaxy and cluster formation. Here we present IRAM
Plateau de Bure Interferometer observations of the enormous Ly$\alpha$ nebulae
`Slug' ($z$=$2.282$) and `Jackpot' ($z$=$2.041$). Our data reveal bright,
synchrotron emission associated with the two radio-loud AGN embedded in the
targeted nebulae, as well as molecular gas, as traced via the CO(3-2) line, in
three galaxies (two sources in the Slug, and one in the Jackpot). All of the CO
emission is associated with galaxies detected in their rest-frame UV stellar
emission. The total mass in molecular gas of these three galaxies [$\sim
(3-5)\times10^{10}$ M$_\odot$] is comparable with the total ionized gas mass
responsible for the diffuse nebular emission. Our observations place limits on
the molecular gas emission in the nebulae: The molecular gas surface density is
$\Sigma_{\rm H2}<12-25$ M$_\odot$ pc$^{-2}$ for the Slug nebula and
$\Sigma_{\rm H2}<34-68$ M$_\odot$ pc$^{-2}$ for the Jackpot nebula. These are
consistent with the expected molecular gas surface densities, as predicted via
photoionization models of the rest-frame UV line emission in the nebulae, and
via Ly$\alpha$ absorption in the Jackpot nebula. Compared to other radio--loud
quasars at $z>1$, and high-redshift radio--loud galaxies, we do not find any
strong trends relating the molecular gas reservoirs, the radio power, and the
Ly$\alpha$ luminosities of these systems. The significant step in sensitivity
required to achieve a detection of the molecular gas from the nebulae, if
present, will require a substantial time investment with JVLA, NOEMA, or ALMA.
