Algebraic isomorphisms of quantized homogeneous spaces

We describe a proof of the following folklore theorem: If $\text{cX}=G\mathrm{/}K$ is the homogeneous space of a simply connected compact semisimple Lie group with Poisson-Lie stabilizers, then the $q$-deformed algebras of regular functions $\text{CC}[{\text{cX}}_q]$ with $0<q\le 1$ are mutually non-isomorphic as $\ast$-algebras.