  Post-starburst galaxies are believed to be in a rapid transition between
major merger starbursts and quiescent ellipticals. Their optical spectrum is
dominated by A-type stars, suggesting a starburst that was quenched recently.
While optical observations suggest little ongoing star formation, some have
been shown to host significant molecular gas reservoirs. This led to the
suggestion that gas depletion is not required to end the starburst, and that
star formation is suppressed by other processes. We present NOEMA CO(1-0)
observations of 15 post-starburst galaxies with emission lines consistent with
Active Galactic Nucleus (AGN) photoionization. We collect post-starburst
candidates with molecular gas measurements from the literature, with some
classified as classical E+A, while others with line ratios consistent with AGN
and/or shock ionization. Using far-infrared observations, we show that systems
that were reported to host exceptionally-large molecular gas reservoirs host in
fact obscured star formation, with some systems showing star formation rates
comparable to ULIRGs. Among E+A galaxies with molecular gas measurements, 7 out
of 26 (26\%) host obscured starbursts. Using far-infrared observations,
post-starburst candidates show similar SFR-$M_{\mathrm{H_2}}$ and
Kennicutt-Schmidt relations to those observed in star-forming and starburst
galaxies. In particular, there is no need to hypothesize star formation
quenching by processes other than the consumption of molecular gas by star
formation. The combination of optical, far-infrared, and CO observations
indicates that some regions within these galaxies have been recently quenched,
while others are still forming stars in highly obscured regions. All this calls
into question the traditional interpretation of such galaxies.
