  The [OIII] 88 $\mu$m fine structure emission line has been detected into the
Epoch of Reionization (EoR) from star-forming galaxies at redshifts $6 < z
\lesssim 9$ with ALMA. These measurements provide valuable information
regarding the properties of the interstellar medium (ISM) in the highest
redshift galaxies discovered thus far. The [OIII] 88 $\mu$m line observations
leave, however, a degeneracy between the gas density and metallicity in these
systems. Here we quantify the prospects for breaking this degeneracy using
future ALMA observations of the [\oiii] 52 $\mu$m line. Among the current set
of ten [OIII] 88 $\mu$m emitters at $6 < z \lesssim 9$, we forecast 52 $\mu$m
detections (at 6-$\sigma$) in SXDF-NB1006-2, B14-6566, J0217-0208, and
J1211-0118 within on-source observing times of 2-10 hours, provided their gas
densities are larger than about $n_{\mathrm{H}} \gtrsim 10^2-10^3$ cm$^{-3}$.
Other targets generally require much longer integration times for a 6-$\sigma$
detection. Either successful detections of the 52 $\mu$m line, or reliable
upper limits, will lead to significantly tighter constraints on ISM parameters.
The forecasted improvements are as large as $\sim 3$ dex in gas density and
$\sim 1$ dex in metallicity for some regions of parameter space. We suggest
SXDF-NB1006-2 as a promising first target for 52 $\mu$m line measurements. We
discuss how such measurements will help in understanding the mass metallicity
relationship during the EoR.
