Davison's stacky PS=WS conjecture

Let $\mathfrak{M}_{Dol}(r,0)$ and $\mathfrak{M}_{B}(r,0)$ be the stacks of semistable Higgs bundles and semisimple representations, respectively. Let $W_\bullet$ be the weight filtration on Borel–Moore homology and $\mathfrak{P}_\bullet$ a suitably defined perverse filtration on the Dolbeault side. Stacky PS=WS conjecture. There exists an isomorphism

$$\Upsilon:H^*_{\mathrm{BM}}(\mathfrak{M}_{B}(r,0),\mathbb{Q})\xrightarrow{\simeq}H^*_{\mathrm{BM}}(\mathfrak{M}_{Dol}(r,0),\mathbb{Q})$$

such that

$$\Upsilon\bigl(W_{2i}H^*_{\mathrm{BM}}(\mathfrak{M}_{B}(r,0),\mathbb{Q})\bigr)=\mathfrak{P}_iH^*_{\mathrm{BM}}(\mathfrak{M}_{Dol}(r,0),\mathbb{Q}).$$

This is the stack-theoretic version of the P=W picture and is formulated in Borel–Moore homology to encompass the stack setting. The source refers to a more precise formulation with shifts, which are omitted in the displayed candidate.