Extend the P=W correspondence to irregular Hitchin systems

Prove that the perverse filtration on the cohomology of type-$A$ wild Hitchin moduli spaces with irregular singularities agrees with the weight filtration of the corresponding mixed Hodge structure on the character variety, thereby extending the $P=W$ conjecture from the regular-singular case to the irregular case.

Background

The paper studies type-$A_{n-1}$ Hitchin moduli spaces on the projective line with one irregular and one regular singularity. It defines two candidate gradings on cohomology: the perverse filtration induced by the Hitchin fibration and the weight filtration of the mixed Hodge structure obtained by viewing the moduli space as a character variety.

For regular singularities, the $P=W$ conjecture predicts an equivalence between these filtrations. The paper explicitly proposes that this equivalence should continue to hold for the wild, irregular systems considered here, but does not establish it.