Poisson structure conjecture for the stack of semistable Higgs bundles over a stacky curve

Let $\mathcal{X}$ be a stacky curve, let $\alpha$ be a fixed parabolic structure, and let $\widetilde{\mathcal{M}}^{ss}_H(\mathcal{X},\alpha)$ denote the moduli stack of semistable Higgs bundles over $\mathcal{X}$ with fixed parabolic structure $\alpha$. Poisson structure conjecture. There is a Poisson structure on

$$\widetilde{\mathcal{M}}^{ss}_H(\mathcal{X},\alpha).$$

The conjecture concerns extending the pointwise Poisson construction to the moduli stack globally; the cited context explains that an atlas by schemes and the universal sheaf on a Quot-scheme may provide the required global construction.