Kaluza-Klein trombone mass matrices and universal class $\mathcal{R}$ operator spectra

We determine universal sectors of the light operator spectrum that are present in all A$_{N-1}$ superconformal field theories $T_N[\Sigma_3]$ of class $\mathcal{R}$, with $\Sigma_3$ a compact hyperbolic three-manifold. We do this at large $N$ by holographically mapping the problem to a Kaluza-Klein spectral analysis over previously known dual anti-de Sitter backgrounds related to wrapped M5-brane configurations. Associated to these, $D=4$ $\mathcal{N}=8$ supergravities have been recently constructed that allow for the application of spectral techniques based on exceptional generalised geometry/field theory. We derive the Kaluza-Klein mass matrices that arise when the seed lower-dimensional maximal supergravity involves gaugings of the trombone scaling symmetry, as pertains to the case at hand, and diagonalise them to find the universal Kaluza-Klein states. In general, these universal spectra are only defined locally, and we give a prescription to isolate globally-defined sectors therein.