[@Maldacena:1997re; @Witten:1998qj] becoming, so far, one of the few known theoretical schemes in which the AdS/CFT correspondence may be explored beyond the supergravity approximation. Furthermore, computations on the AdS$_3$ side can be compared with the two-dimensional CFT description, where powerful techniques are also available, thus giving, in principle, the possibility of constructing dual pairs where both sides can be exactly solved.

Over the last couple of years there has been a renewed interest in this type of AdS/CFT scenario. These efforts are mainly concentrated on the so-called $k=1$ sector, *i.e.* the case with minimal AdS flux. The propagation of strings in space-times of the form AdS$_3\times S^3 \times {\cal{M}}_4$ was considered, including, for example, ${\cal{M}}_4 = T^4$, K3 and $S^3 \times S^1$, and it was argued that the corresponding field theory duals were given by deformations of symmetric orbifold CFT’s [@Gaberdiel:2017oqg; @Eberhardt:2017pty; @Eberhardt:2018ouy; @Eberhardt:2019niq]. Part of these considerations remain valid for the $k>1$ case, albeit now with symmetric orbifold theories containing additional Liouville factors [@Eberhardt:2019qcl].

Unlike the case of a WZNW model having a compact underlying symmetry, when the target space is the universal cover of the SL($2,\mathbb R$) group manifold, the spectrum standardly constructed upon irreducible representations of the 