  We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny
equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the
product of a compact Riemann surface $\Sigma$ with ${\mathbb R}^+_y$, with
generalized Nahm pole boundary conditions at $y=0$. The correspondence is
between solutions of these equations satisfying these singular boundary
conditions and also limiting to flat connections as $y \to \infty$, and certain
holomorphic data consisting of effective triplets $(\mathcal E, \varphi, L)$
where $(\mathcal E, \varphi)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs
pair and $L \subset \mathcal E$ is a holomorphic line bundle. This corroborates
a prediction of Gaiotto and Witten, and is an extension of our earlier paper
\cite{HeMazzeo2017} which treats only the $\mathrm{SL}(2,\mathbb R)$ case.
