Self-duality for $\mathcal{N}$-extended superconformal gauge multiplets

We develop a general formalism of duality rotations for $\mathcal{N}$-extended superconformal gauge multiplets in conformally flat backgrounds as an extension of the approach given in arXiv:2107.02001. Additionally, we construct $\mathsf{U}(1)$ duality-invariant models for the $\mathcal{N}=2$ superconformal gravitino multiplet recently described in arXiv:2305.16029. Each of them is automatically self-dual with respect to a superfield Legendre transformation. A method is proposed to generate such self-dual models, including a family of superconformal theories.