Identify the geometric filtration polynomial represented by the Kostka-weighted parking-function formula

Establish that, for every regular singularity $f$ of a type-$A_{n-1}$ wild Hitchin system with irregularity parameter $m$, the polynomial $C^f_{m,n}(q,t)=\sum_\lambda f_\lambda(q,t)K_{\lambda f}$ agrees with the bigraded polynomial arising from either the mixed Hodge weight filtration or the perverse filtration of the Hitchin fibration.

Background

The paper constructs a combinatorial bigraded polynomial using rational parking functions. For a general regular singularity labeled by a partition $f$, it modifies the Schur coefficients by Kostka numbers, producing $$C^f_{m,n}(q,t)=\sum_\lambda f_\lambda(q,t)K_{\lambda f}$$.

This formula is checked in examples and via affine Springer fiber calculations, but the claimed identification with the actual perverse- or weight-filtered cohomology is left as a conjecture.