  A number of new $z>11$ galaxy candidates have recently been identified based
on public James Webb Space Telescope (JWST) NIRCam observations. Spectroscopic
confirmation of these candidates is necessary to robustly measure their
redshift and put them in the context of our understanding of the buildup of
galaxies in the early Universe. GLASS-z13 is one of these candidates, with a
reported photometric redshift $z>11.9$. I present publicly available Atacama
Large Millimeter/submillimeter Array (ALMA) band 6 Director's Discretionary
Time observations (project 2021.A.00020.S; PI T. Bakx), taken to acquire a
spectroscopic redshift for GLASS-z13 by searching for [OIII] line emission in
the redshift range $z=11.9-13.5$. No [OIII] emission is detected in integrated
spectra extracted within an aperture around GLASS-z13, nor when using an
automated line finding algorithm (applying different uv-weighting strategies
for the imaging). 1.2 mm continuum emission associated to GLASS-z13 is not
detected either. If GLASS-z13 is at z$\approx$12-13, this implies a 3-$\sigma$
upper limit on the [OIII] and rest-frame $\sim$90 $\mu$m continuum emission of
$\sim1\times10^8\,\rm{L}_\odot$ and 10.8 $\mu$Jy, respectively. The
non-detection of [OIII] and continuum emission does not necessarily imply that
GLASS-z13 is not at $z\approx12-13$. It can also be explained by a low
metallicity ($\sim 0.2\,\rm{Z}_\odot$ or lower) and/or high-density (at least
100 $\rm{cm}^{-3}$) interstellar medium. This work demonstrates the synergy
between ALMA and JWST to study the properties of the first galaxies, although
JWST/NIRSpec spectroscopy will be necessary to confirm or reject the high
photometric-redshift of GLASS-z13.
