Is it possible to generalize the Alday-Gaiotto-Tachikawa (AGT) correspondence (duality) to nontrivial geometries?

Given that the origin of the Alday-Gaiotto-Tachikawa (AGT) duality can be traced back to the compactification of the flat space $6$ d $(2,0)$ superconformal field theory J. Yagi, (2012) , I am curious about the possibility of extending this duality to nontrivial geometries. Meaning, if the compactification of $6$ d $(2,0)$ SCFT coupled to gravity A. P. Braun, M. Larfors, P.-K. Oehlmann, (2021) (section 3) [which can be thought of as compactifications of IIB string theory on $K3$ surfaces at specific loci in the moduli space], P. S. Aspinwall, (1996) on a Riemann surface, and search for a duality between the partition function of the $4$ d $\mathcal{N} \geq 2$ SUGRA and a $2$ d CFT similar to that of the Liouville field theory is possible in principle. In particular, one knows that the observables of the $4$ d $\mathcal{N}=2$ SUGRA are already known in terms of the Nekrasov-like partition function K. Hristov , which is quite similar to that of the flat-space $4$ d gauge theory. Finally, such a framework enables one to formally investigate the conjectured existence of a UV fixed point in Einstein's quantum gravity A. B. Platania, (2018)