The absence of both spin freezing and of a static Jahn-Teller effect have lead to the proposition that Ba$_3$CuSb$_2$O$_9$ is a quantum spin-orbital liquid. However, theoretical understanding of the microscopic origin of this behavior has been hampered by a lack of consensus on the lattice structure. Cu ions have been proposed to realize either a triangular lattice, a short-range ordered honeycomb lattice or a disordered lattice with stripelike correlations. Here we analyze the stability of idealiz
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