In a beautiful paper by A. N. Redlich on the parity anomaly (PRL **52**, 18 (1984), no arXiv version), the author indicates that an odd number of Dirac fermions can never be coupled to a massless gauge field in 2+1d due to the necessity of introducing background Chern-Simons terms to maintain gauge invariance

"The induced topological term $±W[A]$ in $I_{eff}[A]$ is known to produce a mass for the gauge fields. Not only must parity conservation be violated in odd-dimensional theories with an odd number of fermions, but the gauge fields $\textit{must}$ become massive as well",

where $W[A]$ is the Chern-Simons term and $I_{eff}[A]$ the effective action obtained by integrating out the fermion degrees of freedom.

My question is whether this is still true if the fermions are at a finite density, i.e. one adds a chemical potential such that there is a whole Fermi surface of excitations (in this case, it's my understanding that one cannot simply integrate out the fermions to get $I_{eff}[A]$). Is it possible to couple an odd number of Dirac fermions, at finite density, to a massless gauge field in 2+1d?

Note: I'm glossing over issues to do with whether the presence of massless fermions can stabilise 2+1d gauge theories against confinement (I'm assuming here that they can). Also I don't mind whether parity conservation is preserved or not, it's just the mass of the gauge field I'm interested in.