I'm recently working on something called *3d mirror symmetry* in QFT literature, which involves two hyperkähler manifolds.
There seems to be a corresponding(?) mathematical theory called *symplectic duality*, pursued by Braden, Licata, Proudfoot and Webster.

Where can I read about it? The only thing I could find so far is the proposal by Proudfoot et al. I'm particularly intrigued by the fifth example in page. 7, which says

More generally, the moduli space of $G$-instantons on a crepant resolution of $\mathbb{C}^2/\Gamma$ is dual to the moduli space of $G'$-instantons on a crepant resolution of $\mathbb{C}^2/\Gamma'$, where $G$ is matched to $\Gamma'$ and $G'$ is matched to $\Gamma$ via the McKay correspondence.

Where can I read about this duality, in particular the case when neither $G$ nor $G'$ is of type $A$?