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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 μm 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 μm 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 $4σ$ 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 $3.8σ$ `[O III] 88 μm' 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.