This is essentially a follow-up question from 'Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?'. Verbitsky's theorem in (https://arxiv.org/pdf/hep-th/9512195.pdf) says that certain compact hyperkaehler manifolds are their own mirrors. What about noncompact hyperkaehler manifolds?
Are the mirrors of noncompact hyperkaehler manifolds also hyperkaehler? Are there at least some examples where this is true? I think at least for noncompact K3 manifolds, this should be known.
In particular, I am interested in noncompact hyperkaehler manifolds which are cotangent bundles. For example, the cotangent bundles of hermitian symmetric spaces are hyperkaehler, as discussed in 'Is the cotangent bundle to a Kahler manifold hyperkahler?'