  The CFT dual of string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times
\mathrm{S}^3 \times \mathrm{S}^1$ is conjectured to be the symmetric orbifold
of the $\mathcal{S}_\kappa$ theory, provided that one of the two $Q_5^\pm$
quantum numbers is a multiple of the other. We determine the BPS spectrum of
the symmetric orbifold in detail, and show that it reproduces precisely the BPS
spectrum that was recently calculated in supergravity. We also determine the
BPS spectrum of the world-sheet theory that is formulated in terms of WZW
models, and show that, apart from some gaps (which are reminiscent of those
that appear in the corresponding $\mathbb{T}^4$ calculation), it also
reproduces this BPS spectrum. In fact, the matching seems to work as well as
for the familiar $\mathbb{T}^4$ case, and thus our results give strong support
for this proposal.
