  We explore how the presence of detectable molecular gas depends on the
inferred star formation histories (SFHs) in 8 massive, quiescent galaxies at
$\mathrm{z\sim0.7}$. Half of the sample have clear detections of molecular gas,
traced by CO(2-1). We find that the molecular gas content is unrelated to the
rate of star formation decline prior to the most recent 1 Gyr, suggesting that
the gas reservoirs are not leftover from their primary star formation epoch.
However, the recent SFHs of CO-detected galaxies demonstrate evidence for
secondary bursts of star formation in their last Gyr. The fraction of stellar
mass formed in these secondary bursts ranges from
$\mathrm{f_{burst}\approx0.3-6\%}$, and ended between
$\mathrm{t_{end\mbox{-}burst}\approx0-330~Myr}$ ago. The CO-detected galaxies
form a higher fraction of mass in the last Gyr
($\mathrm{f_{M_{1Gyr}}=2.6\pm1.8\%}$) compared to the CO-undetected galaxies
($\mathrm{f_{M_{1Gyr}}=0.2\pm0.1\%}$). The galaxies with gas reservoirs have
enhanced late-time star formation, highlighting this as a contributing factor
to the observed heterogeneity in the gas reservoirs in high-redshift quiescent
galaxies. We find that the amount of gas and star formation driven by these
secondary bursts are inconsistent with that expected from dry minor mergers,
and instead are likely driven by recently-accreted gas i.e., gas-rich minor
mergers. This conclusion would not have been made based on
$\mathrm{SFR_{UV+IR}}$ measurements alone, highlighting the power of detailed
SFH modeling in the interpretation of gas reservoirs. Larger samples are needed
to understand the frequency of low-level rejuvenation among quiescent galaxies
at intermediate redshifts, and to what extent this drives the diversity of
molecular gas reservoirs.
