Mirror symmetry for the Painlev\'e character varieties

We establish a homological mirror theorem for the 4-manifolds arising as moduli of (irregular) rank two local systems on the projective line. Specifically, we prove that the Fukaya category of a moduli of such local systems with generic microlocal monodromy at punctures is equivalent to the category of coherent sheaves on the minimal resolution of the corresponding moduli of local systems with trivial microlocal monodromy.