  The claim of an exomoon candidate in the Kepler-1625b system has generated
substantial discussion regarding possible alternative explanations for the
purported signal. In this work we examine in detail these possibilities. First,
the effect of more flexible trend models is explored and we show that
sufficiently flexible models are capable of attenuating the signal, although
this is an expected byproduct of invoking such models. We also explore trend
models using X and Y centroid positions and show that there is no data-driven
impetus to adopt such models over temporal ones. We quantify the probability
that the 500 ppm moon-like dip could be caused by a Neptune-sized transiting
planet to be < 0.75%. We show that neither autocorrelation, Gaussian processes
nor a Lomb-Scargle periodogram are able to recover a stellar rotation period,
demonstrating that K1625 is a quiet star with periodic behavior < 200 ppm.
Through injection and recovery tests, we find that the star does not exhibit a
tendency to introduce false-positive dip-like features above that of pure
Gaussian noise. Finally, we address a recent re-analysis by Kreidberg et al
(2019) and show that the difference in conclusions is not from differing
systematics models but rather the reduction itself. We show that their
reduction exhibits i) slightly higher intra-orbit and post-fit residual
scatter, ii) $\simeq$ 900 ppm larger flux offset at the visit change, iii)
$\simeq$ 2 times larger Y-centroid variations, and iv) $\simeq$ 3.5 times
stronger flux-centroid correlation coefficient than the original analysis.
These points could be explained by larger systematics in their reduction,
potentially impacting their conclusions.
