Twisted parabolic Higgs-moduli perverse Hodge polynomial conjecture

Let $C$ be a complex smooth projective curve of genus $g$, with divisor $D=p_1+\dots+p_k+rp$, where the points are distinct and $p$ has multiplicity $r$. Let ${\bm \mu}$ be a $k$-tuple of partitions, let ${\mathcal{M}}_\Dol^{{\bm \mu},r}$ be the associated smooth moduli space of stable parabolic Higgs bundles, and let $d_{{\bm \mu},r}$ be its dimension. Its perverse Hodge polynomial is

$$PH({\mathcal{M}}_\Dol^{{\bm \mu},r};q,t)=\sum \dim\left(\operatorname{Gr}^P_i(H^k({\mathcal{M}}_\Dol^{{\bm \mu},r}))\right)q^it^k.$$

Let $\mathbb{H}_{{\bm \mu},r}(z,w)$ denote the polynomial defined by the paper. Perverse Hodge polynomial conjecture. We expect

$$PH({\mathcal{M}}_\Dol^{{\bm \mu},r};q,t)=(qt^2)^{d_{{\bm \mu},{\bf r}}}\mathbb{H}_{{\bm \mu},r}(q^{-1/2},-q^{1/2}t).$$

The cited string-theoretical work gives evidence for this identity, while the source presents it as a mathematical conjecture rather than a proved result.