Difference formula for tensor-power multiplicities of covariant representations of $\mathfrak{sl}(m|n)$

Let $\mathfrak{sl}(m|n)$ be the type $A$ Lie superalgebra, let $V_{(2s)}$ be its covariant representation, and write

$$V_{(2s)}^{\otimes L}=\bigoplus_{\lambda}\mu_{\lambda}V_{\lambda},$$

where $\lambda$ ranges over $(m,n)$-hook Young diagrams. Decompose $\lambda$ into its first $m$ rows and the remaining diagram $\tilde{\lambda}$, and let $\vec M=(M_0,\ldots,M_{m+n})$ satisfy $M_0=2sL$, $M_{m+n}=0$,

$$\lambda_i=M_{i-1}-M_i,\qquad \tilde{\lambda}^{\prime}_{\ell}=M_{\ell+m-1}-M_{\ell+m}.$$

Here $\tilde{\lambda}^{\prime}$ is the conjugate Young diagram of $\tilde{\lambda}$, with $i=1,\ldots,m$ and $\ell=1,\ldots,n$. The tensor-power difference formula. Under this correspondence,

$$\mu_{\lambda}=\mathcal{D}_{R_{\mathfrak{sl}(m|n)}(t)}c_{s,L}(\vec M),$$

where $\mathcal{D}_{R_{\mathfrak{sl}(m|n)}(t)}$ is the shifted operator associated with $R_{\mathfrak{sl}(m|n)}(t)$. This conjectures a superalgebraic analogue of the difference formula relating weight-space coefficients to tensor-product multiplicities; its status is not resolved in the supplied text.