  The molecular gas content of high-redshift galaxies is a highly sought-after
property. However, H$_2$ is not directly observable in most environments, so
its mass is probed through other emission lines (e.g., CO, [CI], [CII]), or
through a gas-to-dust ratio. Each of these methods depends on several
assumptions, and are best used in parallel. In this work, we extend an
additional molecular gas tracer to high-redshift studies by observing hydrogen
deuteride (HD) emission in the strongly lensed $z=5.656$ galaxy SPT0346-52 with
ALMA. While no HD(1-0) emission is detected, we are able to place an upper
limit on the gas mass of $\rm M_{H_2}<6.4\times10^{11} M_{\odot}$. This is used
to find a limit on the $\rm L'_{CO}$ conversion factor of $\rm\alpha_{CO}<5.8$
M$_{\odot}$(K km s$^{-1}$ pc$^2$)$^{-1}$. In addition, we construct the most
complete spectral energy distribution (SED) of this source to date, and fit it
with a single-temperature modified blackbody using the nested sampling code
MultiNest, yielding a best-fit dust mass $\rm M_{dust}=10^{8.92\pm0.02}$
M$_{\odot}$, dust temperature $78.6\pm0.5$ K, dust emissivity spectral index
$\beta=1.81\pm0.03$, and star formation rate $\rm SFR=3800\pm100$ M$_{\odot}$
year$^{-1}$. Using the continuum flux densities to estimate the total gas mass
of the source, we find $\rm M_{H_2}<2.4\times10^{11}$ M$_{\odot}$, assuming
sub-solar metallicity. This implies a CO conversion factor of $\rm
\alpha_{CO}<2.2$, which is between the standard values for MW-like galaxies and
starbursts. These properties confirm that SPT0346-52 is a heavily starbursting,
gas rich galaxy.
