The PI=WI conjecture for singular character varieties

Let $r,n$ be integers with $r>0$, not necessarily coprime, and let $\mathcal{M}_{B}(r,n)$ and $\mathcal{M}_{Dol}(r,n)$ be the Betti and Dolbeault moduli spaces. Let $\psi:\mathcal{M}_{B}(r,n)\to\mathcal{M}_{Dol}(r,n)$ be the real analytic nonabelian Hodge isomorphism, and let $P_\bullet$ and $W_\bullet$ denote the perverse and weight filtrations on intersection cohomology. PI=WI conjecture. The induced isomorphism

$$\psi^*:IH^*(\mathcal{M}_{Dol}(r,n),\mathbb{Q})\to IH^*(\mathcal{M}_{B}(r,n),\mathbb{Q})$$

should satisfy, for every $k\in\mathbb{Z}$,

$$P_kIH^*(\mathcal{M}_{Dol}(r,n),\mathbb{Q})\xrightarrow{\simeq}W_{2k}IH^*(\mathcal{M}_{B}(r,n),\mathbb{Q}).$$

This is the proposed singular analogue of P=W, replacing ordinary cohomology by intersection cohomology because the moduli spaces are generally singular in the noncoprime case. The source cites evidence from complex-structure independence of the perverse filtration, but gives no resolution of the conjecture.