The equivariant Brauer groups of commuting free and proper actions are isomorphic

If $X$ is a locally compact space which admits commuting free and proper actions of locally compact groups $G$ and $H$, then the Brauer groups ${\text{Br}}_H(G\mathrm{/}X)$ and ${\text{Br}}_G(X\mathrm{/}H)$ are naturally isomorphic.