  It has been recently claimed that KOIs-268.01, 303.01, 1888.01, 1925.01,
2728.01 & 3320.01 are exomoon candidates, based on an analysis of their transit
timing. Here, we perform an independent investigation, which is framed in terms
of three questions: 1) Are there significant excess TTVs? 2) Is there a
significant periodic TTV? 3) Is there evidence for a non-zero moon mass? We
applied rigorous statistical methods to these questions alongside a re-analysis
of the Kepler photometry and find that none of the KOIs satisfy these three
tests. Specifically, KOIs-268.01 & 3220.01 pass none of the tests and
KOIs-303.01, 1888.01 & 1925.01 pass a single test each. Only KOI-2728.01
satisfies two, but fails the cross-validation test for predictions. Further,
detailed photodynamical modeling reveals that KOI-2728.01 favours a negative
radius moon (as does KOI-268.01). We also note that we find a significant
photoeccentric for KOI-1925.01 indicating an eccentric orbit of
e>(0.62+/-0.06). For comparison, we applied the same tests to Kepler-1625b,
which reveals that 1) and 3) are passed, but 2) cannot be checked with the
cross-validation method used here, due to the limited number of available
epochs. In conclusion, we find no compelling evidence for exomoons amongst the
six KOIs. Despite this, we're able to derive exomoon mass upper limits versus
semi-major axis, with KOI-3220.01 leading to particularly impressive
constraints of Ms/Mp < 0.4% [2 sigma] at a similar relative semi-major to that
of the Earth-Moon.
