  Determining when the first galaxies formed remains an outstanding goal of
modern observational astronomy. Theory and current stellar population models
imply that the first galaxies formed at least at $z = 14-15$. But to date, only
one galaxy at $z > 13$ (GS-z13-0) has been spectroscopically confirmed.. The
galaxy `HD1' was recently proposed to be a z=13.27 galaxy based on its
potential Lyman break and tentative [O III] 88 {\mu}m detection with ALMA. We
hereby aim to test this scenario with new ALMA Band 4, DDT observations of what
would be the [C II] 158 {\mu}m emission, if HD1 is at z$\sim$13.27. We
carefully analyse the new ALMA Band 4 observations and re-analyse the existing
ALMA Band 6 data on the source to determine the proposed redshift. We find a
tentative $4\sigma$ feature in the Band 4 data that is spatially offset by 1.7"
and spectrally offset by 190 km s-1 from the previously-reported $3.8\sigma$
`[O III] 88 {\mu}m' feature. Through various statistical tests, we demonstrate
that these tentative features are fully consistent with both being random noise
features. We conclude that we are more likely to be recovering noise features
than both [O III] and [C II] emission from a source at $z\sim 13.27$. Although
we find no credible evidence of a $z\sim 13.27$ galaxy, we cannot entirely rule
out this scenario. Non-detections are also possible for a $z\sim 13$ source
with a low interstellar gas-phase metallicity or ionisation parameter and/or
high gas density. Determining where and exactly what type of galaxy HD1 is,
will now likely require JWST/NIRSpec spectroscopy.
