The wild character variety and P = W conjecture for plane curve singularities

Let $C$ be a locally planar rational curve with a unique singularity, and let $\Lambda$ be a rainbow closure of a positive braid such that the topological knot underlying $\Lambda$ is the link of the singularity. Let $\mathcal{M}_1(\Lambda)$ be the corresponding rank-one constructible-sheaf moduli space, and let $\overline{J}(C)$ be the compactified Jacobian of $C$. Wild character variety and P = W conjecture. The space $\mathcal{M}_1(\Lambda)$ should be the wild character variety that corresponds, under the nonabelian Hodge theorem, to a wild Hitchin system whose central fibre is $\overline{J}(C)$; moreover, $P = W$ should hold in this setting. This conjecturally gives a conceptual explanation of the preceding equality between the perverse filtration on the compactified Jacobian and the weight filtration on the constructible-sheaf moduli space; the source states no resolution.