  Let $\mathfrak{g}$ be a complex finite-dimensional semisimple Lie algebra and
$\mathfrak{k}$ be any $\mathrm{sl}(2)$-subalgebra of $\mathfrak{g}$. In this
paper we prove an earlier conjecture by Penkov and Zuckerman claiming that the
first derived Zuckerman functor provides an equivalence between a truncation of
a thick parabolic category $\mathcal{O}$ for $\mathfrak{g}$ and a truncation of
the category of admissible $(\mathfrak{g}, \mathfrak{k})-$modules. This latter
truncated category consists of admissible $(\mathfrak{g},
\mathfrak{k})-$modules with sufficiently large minimal $\mathfrak{k}$-type. We
construct an explicit functor inverse to the Zuckerman functor in this setting.
As a corollary we obtain an estimate for the global injective dimension of the
inductive completion of the truncated category of admissible $(\mathfrak{g},
\mathfrak{k})-$modules.
