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Enormous Ly$α$ 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$α$ 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 $Σ_{\rm H2}<12-25$ M$_\odot$ pc$^{-2}$ for the Slug nebula and $Σ_{\rm H2}<34-68$ 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$α$ 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$α$ 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.