SQuIGG$\vec{L}$E: Spatially resolved NIRSpec Pa$α$ reveals that the most CO-luminous, $z\sim0.7$ post-starburst galaxies exhibit falling---but still active---star formation behind optically thick dust
One of the fundamental uncertainties in the study of galaxies at the interface between star formation and quiescence--post-starburst galaxies--is their instantaneous star formation rate. While these galaxies appear quiescent in the rest UV-optical, many of the youngest systems are highly CO luminous, implying dense molecular gas, and potentially star formation. However, their instantaneous star formation rates have proved elusive, as measurements with rest-optical emission lines or UV-to-IR SED fitting yield order-of-magnitude uncertainty. Here, we present JWST/NIRSpec G395M/170LP integral field spectroscopy of the three most CO-luminous massive galaxies in the SQuIGG$\vec{L}$E survey, spatially resolving Pa$α$/Br$γ$/Br$β$ to map the dust-corrected star formation rate. We find that these galaxies host star formation rates of $27-80$ $M_\odot \ \mathrm{yr^{-1}}$, though $\sim30-40\%$ of the emission may arise from non-star-forming ionization (AGN or shocks), implied by high [FeII]/Pa$α$ in central spaxels. We also find effective $A_{\mathrm{Pa}α}\sim1$ and individual $\sim700$ pc spaxels with $A_{\mathrm{Paα}}\sim2$, implying significant dust obscuration of nebular regions that contribute little to the rest-optical lines. These galaxies are not consistent with total quiescence; the measured star formation rates place these galaxies on or above the main sequence. However, their star formation rates \textit{are} significantly lower ($\sim0.5-1$ dex) than the peak star formation rates inferred from spectrophotometric fitting. We propose that the CO luminous post-starburst phase represents the last gasp of star formation in galaxies on the rapid quenching pathway, and that dust-obscured star formation, outflows, and (speculatively) an uncertain CO-to-H2 conversion reconcile their gas exhaustion with observed timescales of fading.