Expected class identity for symmetric powers of twisted projective spaces

Let $\mathcal{E}$ be an $\alpha$-twisted locally free sheaf on $\operatorname{Spec}K$ of rank $n$, and let $\mathbb{P}\mathcal{E}$ denote its associated twisted projective space. The expected class identity. For every $k\geq 0$, one should have

$$[\operatorname{Sym}^{nk+1}(\mathbb{P}\mathcal{E})]=[\operatorname{Gr}^{nk+1}(\mathcal{E}^{\oplus(k+1)})].$$

This is presented as a natural expectation extending the class equality between symmetric powers of projective space and Grassmannians. The source gives no evidence that it has been proved or disproved.