Weight-filtration and hard-Lefschetz conjecture for affine Toda centralizers

Let $G$ be a semisimple group over $\mathbb{C}$, let $G^{\mathrm{sc}}$ be its simply-connected cover, and let $J^G_{G^{\mathrm{sc}}}$ be the associated regular centralizer group scheme. Write $r$ for the rank of $G$, let $\widetilde{I}_d$ be the set of affine simple roots whose Dynkin labeling is divisible by $d$, let $r_d=|\widetilde{I}_d|$, and let $\varphi$ be Euler's totient function. Weight-filtration and hard-Lefschetz conjecture. The following hold: $\operatorname{H}^{i}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})=0$ for odd $i$; for $0\leq j\leq r$ and $j\leq i\leq 2j$,

$$\dim \operatorname{Gr}^{W}_{2i}\operatorname{H}^{2j}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})=\sum_{d\in\mathbb{N},\,\widetilde{I}_d\neq\varnothing,\,2j-i=r-r_d+1}\varphi(d),$$

and otherwise $\operatorname{Gr}^{W}_{2i}\operatorname{H}^{2j}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})=0$. In particular, $\dim\operatorname{H}^{2}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})=1$ and it is pure of weight $4$. If $\eta\in\operatorname{H}^{2}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})$ is nonzero, then cupping with $\eta^i$ induces an isomorphism

$$\cup\eta^i:\operatorname{Gr}^{W}_{2r-2i}\operatorname{H}^{*}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})\stackrel{\sim}{\longrightarrow}\operatorname{Gr}^{W}_{2r+2i}\operatorname{H}^{*+2i}(J^G_{G^{\mathrm{sc}}},\mathbb{Q})$$

for $0\leq i\leq r$.

These predictions refine the P=W conjecture and the computed cohomology of $\mathcal{M}^{\circ}_G$, specifying the expected weights and a hard-Lefschetz pattern on the centralizer side. The source does not state that this conjecture has been resolved.