Semisimplicity preservation for Harish-Chandra modules

Let $\mathcal{H}(\chi)_{\mathbb O}$ be the Harish-Chandra category associated with a nilpotent orbit $\mathbb O$, and let ${}^\chi \mathcal{H}^K_{\mathbb O}$ be the full subcategory of semisimple objects in the corresponding quotient category of $(\mathfrak{g},K)$-modules. For $M\in \mathcal{H}(\chi)_{\mathbb O}$ and $N\in {}^\chi \mathcal{H}^K_{\mathbb O}$, the tensor product $M\otimes N$ belongs to ${}^\chi \mathcal{H}^K_{\mathbb O}$. Semisimplicity-preservation conjecture. The action of $\mathcal{H}(\chi)_{\mathbb O}$ on the quotient category preserves its full subcategory of semisimple objects. This would extend the multi-fusion-category action from Harish-Chandra bimodules to Harish-Chandra $(\mathfrak{g},K)$-modules; the statement is presented as a conjectural direction, and no resolution is given here.