  Exomoons represent an outstanding challenge in modern astronomy, with the
potential to provide rich insights into planet formation theory and
habitability. In this work, we stack the phase-folded transits of 284 viable
moon hosting Kepler planetary candidates, in order to search for satellites.
These planets range from Earth-to-Jupiter sized and from ${\sim}$0.1-to-1.0 AU
in separation - so-called "warm" planets. Our data processing includes two-pass
harmonic detrending, transit timing variations, model selection and careful
data quality vetting to produce a grand light curve with an r.m.s. of 5.1 ppm.
We find that the occurrence rate of Galilean-analog moon systems for planets
orbiting between ${\sim}$0.1 and 1.0 AU can be constrained to be $\eta<0.38$ to
95% confidence for the 284 KOIs considered, with a 68.3% confidence interval of
$\eta=0.16_{-0.10}^{+0.13}$. A single-moon model of variable size and
separation locates a slight preference for a population of short-period moons
with radii ${\sim}0.5$ $R_{\oplus}$ orbiting at 5-10 planetary radii. However,
we stress that the low Bayes factor of just 2 in this region means it should be
treated as no more than a hint at this time. Splitting our data into various
physically-motivated subsets reveals no strong signal. The dearth of
Galilean-analogs around warm planets places the first strong constraint on
exomoon formation models to date. Finally, we report evidence for an exomoon
candidate Kepler-1625b I, which we briefly describe ahead of scheduled
observations of the target with the Hubble Space Telescope.
