The Read-inspired superalgebra isomorphism for the spin Weyl group of type D

Let $n$ be even. The Clifford algebra ${\mathcal Cl}_n$ is generated by $n$ odd generators, and $\mathbb C A_n$ is regarded as an even algebra. The spin group algebra is $\mathbb C D_n^-$.

Read-inspired isomorphism conjecture. There is a superalgebra isomorphism

$${\mathcal Cl}_n \otimes \mathbb C A_n \cong \mathbb C D_n^-.$$

This proposed structure result is motivated by Read's work and by the equality between the number of split classes of $\mathbb C D_n^-$ and the number of conjugacy classes of the alternating group $A_n$. The supplied text does not establish the isomorphism, so its resolution remains open.