  We consider the $\mathcal{N}=(2,2)$ AdS$_3$/CFT$_2$ dualities proposed by
Eberhardt, where the bulk geometry is AdS$_3\times(S^3\times
T^4)/\mathbb{Z}_k$, and the CFT is a deformation of the symmetric orbifold of
the supersymmetric sigma model $T^4/\mathbb{Z}_k$ (with $k=2,\ 3,\ 4,\ 6$). The
elliptic genera of the two sides vanish due to fermionic zero modes, so for
microstate counting applications one must consider modified supersymmetric
indices. In an analysis similar to that of Maldacena, Moore, and Strominger for
the standard $\mathcal{N}=(4,4)$ case of $T^4$, we study the appropriate
helicity-trace index of the boundary CFTs. We encounter a strange phenomenon
where a saddle-point analysis of our indices reproduces only a fraction
(respectively $\frac{1}{2},\ \frac{2}{3},\ \frac{3}{4},\ \frac{5}{6}$) of the
Bekenstein-Hawking entropy of the associated macroscopic black branes.
