Relative Langlands duality of the Bump-Friedberg-Ginzburg ${\mathrm{G}\mathrm{S}\mathrm{O}}_6$-integral

We provide a new instance of singular relative Langlands duality, underlying a Rankin-Selberg integral on ${\mathrm{G}\mathrm{S}\mathrm{O}}_6$ due to Bump-Friedberg-Ginzburg. We conclude that this integral represents an essentially self-dual object in the relative Langlands program, and we demonstrate that the Langlands dual automorphic integral computes a finite sum of $L$-functions, reflecting stacky structure on the spectral side.