  Under broad hypotheses we derive a scalar reduction of the generalized
K\"ahler-Ricci soliton system. We realize solutions as critical points of a
functional analogous to the classical Aubin energy defined on the orbit of a
natural Hamiltonian action of diffeomorphisms, thought of as a generalized
K\"ahler class. This functional is convex on a large set of paths in this
space, and using this we show rigidity of solitons in their generalized
K\"ahler class. As an application we prove uniqueness of the generalized
K\"ahler-Ricci solitons on Hopf surfaces constructed in arXiv:1907.03819,
finishing the classification in complex dimension $2$.
