Mirror symmetry conjecture for categories of line operators

Let $\calT$ be a three-dimensional $N=4$ theory and let $\calT^*$ denote its mirror. Write $\calC_C(\calT)$ and $\calC_H(\calT)$ for the Coulomb and Higgs categories of line operators, with canonical objects $\calF_C(\calT)$ and $\calF_H(\calT)$. Mirror symmetry conjecture. The category $\calC_C(\calT)$ is equivalent to $\calC_H(\calT^*)$, and likewise with $C$ and $H$ interchanged; under this equivalence $\calF_C(\calT)$ maps to $\calF_H(\calT^*)$. This is the categorical formulation of three-dimensional mirror symmetry and predicts an exchange of Coulomb- and Higgs-branch structures.