P=W conjecture for affine Toda moduli spaces

Let $G$ be a semisimple group over $\mathbb{C}$, let $\mathcal{M}^{\circ}_G$ be the neutral component of the affine Toda moduli space, and let $J^G_{G^{\mathrm{sc}}}$ be the associated regular centralizer group scheme. Let $P_i$ denote the perverse filtration on $\operatorname{H}^{*}(\mathcal{M}^{\circ}_G,\mathbb{Q})$ associated to the Hitchin fibration, and let $W_{2i}$ denote the weight filtration on $\operatorname{H}^{*}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})$. P=W conjecture. Under the homeomorphism $\mathcal{M}^{\circ}_G\simeq J^G_{G^{\mathrm{sc}}}$ in the hyperKähler rotation conjecture, the induced isomorphism

$$\operatorname{H}^{*}(\mathcal{M}^{\circ}_G,\mathbb{Q})\stackrel{\sim}{\longrightarrow}\operatorname{H}^{*}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})$$

sends $P_i\operatorname{H}^{*}(\mathcal{M}^{\circ}_G,\mathbb{Q})$ isomorphically to $W_{2i}\operatorname{H}^{*}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})$ for $i=0,1,\ldots,2r$.

This is the affine-Toda analogue of the P=W conjecture for the usual Hitchin moduli space and character variety. It depends on the preceding hyperKähler rotation conjecture, and the source does not state that either conjecture has been resolved.