  We consider $\mathcal{N}=(2,2)$ AdS$_3$/CFT$_2$ dualities proposed in the
large central charge limit ($c\to\infty$) by Eberhardt. Here we propose the
associated D1-D5 systems to be orbifolds of the standard $\mathcal{N}=(4,4)$
systems, thereby elevating the dualities to the finite-$c$ level on the
boundary and to the quantum level in the bulk. In particular, we show that our
brane systems yield low-energy sigma models whose subleading central charges
match earlier predictions from bulk one-loop supergravity computations. In the
case involving the Enriques surface, the finite-$c$ sigma model has a
non-trivial elliptic genus which we use to microscopically explain both the
Bekenstein-Hawking entropy and the subleading logarithmic correction to it for
the associated macroscopic black brane.
