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In experiment, synthesis often is not performed at fixed stoichiometry, but rather in an atmosphere with, for example, tunable oxygen pressure. To capture this configuration, we model the La-Cu-S system open to oxygen, which is tuned by a chemical potential $\mu(\text{O}_2)$, using the formalism developed in \cite{Ping_Ong_2008}. The stability energies of our two target compounds, along with LaCuSO, are shown as a function of the oxygen potential in Fig.~\ref{fig:mu-lcso}. We find that enhancing the oxygen potential does not help stabilize our target compounds. Likely, this is because the underlying cause is that the flexible valence of sulfur effectively reduces copper from its starting 2+ oxidation state. Copper tends to be more stable in its 1+ state, and thus forms LaCuSO instead of our target compounds, which require a 2+ copper state.  If we had access to this information during the original work, we would not have recommended that experimental groups attempt to synthesize the target compounds because instability energies of over 0.5eV/atom are quite large. If instead the energies lay up $\sim$100meV/atom above the hull, we would believe there is a reasonable probability the new composition would be stable. Thus, notwithstanding the systematic uncertainties in choice of $U$ and correction parameters, convex hulls are critical to evaluating potential new compounds.