Reduction conjecture for $GL(m|n)$ tensor categories

Let $m\geq n$, let $\mathcal{T}_{m|n}^+$ be the general tensor category, and let $L(\lambda)$ be an irreducible representation with $\operatorname{sdim}(L(\lambda))>0$. Let $L(\lambda^0)$ be its image under the standard block equivalence with the principal block of $\mathcal{T}_{n|n}$, and write $H_{\lambda^0}$ for the corresponding group. Reduction conjecture for $GL(m|n)$.

$$H_{\lambda}\cong \operatorname{Rep}(GL(m-n))\times H_{\lambda^0},$$

and $L(\lambda)$ corresponds to $L_{\Gamma}\boxtimes V_{\lambda^0}$, where $L_{\Gamma}$ is an irreducible representation of $GL(m-n)$ depending only on the block $\Gamma$. The claim would extend the $m=n$ description to the general $GL(m|n)$ case.