A tentative proposal towards an equivariant mirror symmetry for Hitchin systems

Motivated by Aganagic's equivariant mirror symmetry for certain Coulomb branches of a $3d$ $\mathcal{N}= 4$ gauge quiver theory, we would like to propose a set of ideas towards an extension of Aganagic's proposal to Hitchin systems. At the end, there are two main points in our proposal; namely, that the equivariant mirror of the Hitchin systems should be a Landau-Ginzburg model (with twisted masses) and that the dichotomy between additive and multiplicative varieties in the context of mirror symmetry for Nakajima quiver varieties should be considered in the case of Hitchin systems.