SQuIGG$\vec{L}$E: Buried star formation cannot explain the rapidly fading CO(2-1) luminosity in massive, $z\sim0.7$ post-starburst galaxies

Observational and theoretical studies have long held that rapid gas consumption in starbursts is responsible for the formation of quiescent galaxies. However, studies of recently quenched ``post-starburst" galaxies have discovered that a number of them are surprisingly luminous in CO, challenging this assumption. We present deep ALMA CO(2-1) observations of 50 massive ($\log(M_\star/M_\odot)\sim11.2$) post-starburst galaxies from the SQuIGG$\vec{L}$E sample at $z\sim0.7$. We detect a large fraction (27/50) of the galaxies in CO(2-1). Furthermore, we find that the CO luminosity correlates with the age of the recent starburst, suggesting a gas-removal timescale of $\lesssim140$ Myr, an order of magnitude shorter than is implied by their rest optical star formation rates. We perform new spectral energy distribution fits incorporating mid- and far-IR photometry to test whether dust-obscured star formation can explain this trend. We find that while allowing for buried star formation can raise star formation rates by $\sim0.5$ dex, for almost all galaxies it is neither required to fit the observed IR SED, nor is it sufficient to explain the observed depletion trend. Even the combination of significant buried star formation and ULIRG-like $\alpha_{CO}$ is not enough to explain this decay in CO luminosity. Furthermore, there is no strong evidence to support either of those modifications to the depletion time. Therefore, it remains a distinct possibility that the age-CO luminosity trend should not be interpreted as an evolutionary sequence, and that gas-rich SQuIGG$\vec{L}$E galaxies will soon rejuvenate.