  Kepler and Hubble photometry of a total of four transits by the Jupiter-sized
Kepler-1625b have recently been interpreted to show evidence of a Neptune-sized
exomoon. The profound implications of this first possible exomoon detection and
the physical oddity of the proposed moon, that is, its giant radius prompt us
to re-examine the data and the Bayesian Information Criterion (BIC) used for
detection. We combine the Kepler data with the previously published Hubble
light curve. In an alternative approach, we perform a synchronous polynomial
detrending and fitting of the Kepler data combined with our own extraction of
the Hubble photometry. We generate five million MCMC realizations of the data
with both a planet-only model and a planet-moon model and compute the BIC
difference (DeltaBIC) between the most likely models, respectively. DeltaBIC
values of -44.5 (using previously published Hubble data) and -31.0 (using our
own detrending) yield strongly support the exomoon interpretation. Most of our
orbital realizations, however, are very different from the best-fit solutions,
suggesting that the likelihood function that best describes the data is
non-Gaussian. We measure a 73.7min early arrival of Kepler-1625b for its Hubble
transit at the 3 sigma level, possibly caused by a 1 day data gap near the
first Kepler transit, stellar activity, or unknown systematics. The radial
velocity amplitude of a possible unseen hot Jupiter causing Kepler-1625b's
transit timing variation could be some 100m/s. Although we find a similar
solution to the planet-moon model as previously proposed, careful consideration
of its statistical evidence leads us to believe that this is not a secure
exomoon detection. Unknown systematic errors in the Kepler/Hubble data make the
DeltaBIC an unreliable metric for an exomoon search around Kepler-1625b,
allowing for alternative interpretations of the signal.
