Cohomological induction equivalence conjecture for generalized Harish-Chandra modules

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra, let $\mathfrak{k}\simeq\mathfrak{sl}(2)$ be a subalgebra, and let $\mathfrak{t}$ be a Cartan subalgebra of $\mathfrak{k}$. For $n\geq 0$, let $\mathcal{C}_{\bar{\mathfrak{p}},\mathfrak{t},n}$ be the full subcategory of $\mathfrak{g}$-modules consisting of finite-length modules whose simple subquotients are $\bar{\mathfrak{p}}$-locally finite $(\mathfrak{g},\mathfrak{t})$-modules with $\mathfrak{t}$-weight spaces $N^\beta$, $\beta\in\mathbb{Z}$, satisfying $\beta\geq n$. Let $\mathcal{C}_{\mathfrak{k},n}$ be the full subcategory of $\mathfrak{g}$-modules consisting of finite-length modules whose simple subquotients are $(\mathfrak{g},\mathfrak{k})$-modules with minimal $\mathfrak{k}$-type $V(\mu)$ for $\mu\geq n$. Cohomological induction equivalence conjecture. If $n\geq\Lambda$, then

$$R^{1}\Gamma_{\mathfrak{k},\mathfrak{t}}:\mathcal{C}_{\bar{\mathfrak{p}},\mathfrak{t},n+2}\longrightarrow\mathcal{C}_{\mathfrak{k},n}$$

is an equivalence between the categories $\mathcal{C}_{\bar{\mathfrak{p}},\mathfrak{t},n+2}$ and $\mathcal{C}_{\mathfrak{k},n}$. Here $\Lambda$ is the bound for the genericity condition associated with the pair $(\mathfrak{g},\mathfrak{k})$. The conjecture strengthens the known result that $R^{1}\Gamma_{\mathfrak{k},\mathfrak{t}}$ is fully faithful from $\mathcal{C}_{\bar{\mathfrak{p}},\mathfrak{t},n+2}$ to $\mathcal{C}_{\mathfrak{k},n}$ for $n\geq0$; the source does not state whether essential surjectivity, and hence the equivalence, has been proved.