$C(S{O}_q(2n+1)\mathrm{/}S{O}_q(2n-1))$ as iterated torsioned quantum double suspensions of $C(\mathbb{T})$

Let $A$ be a unital $C^{\ast }$-algebra, and let ${\mathrm{\Sigma }}_m^{2}A$ denote the $m$-torsioned quantum double suspension of $A$. For $q\in (0,1)$ and $n\ge 1$, we prove that the $C^{\ast }$-algebra corresponding to the quotient space $S{O}_q(2n+1)\mathrm{/}S{O}_q(2n-1)$ is isomorphic to ${\mathrm{\Sigma }}^{2(n-1)}{\mathrm{\Sigma }}_2^2{\mathrm{\Sigma }}^{2(n-1)}C(\mathbb{T})$. It follows as a consequence that these spaces are independent of the deformation parameter $q$.