Folded skew Howe duality for spin centralizer algebras

Let $n\geq 1$ and $m\geq 2$. Write $K_0^\tau(-)$ for the weighted Grothendieck group of a category with involution, let $({\EuScript{B}}_{m}^{n})^\tau$ be the equivariant category, and let $\mathcal{F}_i^{(k)}\mathcal{E}_i^{(k)}\mathbbm{1}_{\mathbf n}$ and $\mathsf{X}_i^{(k)}$ denote the corresponding categorical and endomorphism-algebra elements. Folded skew Howe duality. There is an isomorphism of ${\mathbb C}(q)$-algebras

$${\mathbb C}(q)\otimes_{{\mathbb Z}[q^\pm]}K_0^\tau(({{\EuScript{B}}_{m}^{n}})^\tau)\xrightarrow{\cong}{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes m})$$

that sends the class of an appropriate equivariant structure on $\mathcal{F}_i^{(k)}\mathcal{E}_i^{(k)}\mathbbm{1}_{\mathbf n}$ to $\mathsf{X}_i^{(k)}$. This generalizes the established $m=2$ case and proposes an equivariant categorification of the spin centralizer algebra for arbitrary tensor powers.