Full support conjecture for the degree-zero Hitchin system

Let $M(n,0)$ be the degree-zero moduli space in the Hitchin fibration, with Hitchin map $\chi(n,0)$ to its basis. The complex $R\chi(n,0)_*\operatorname{IC}(M(n,0),\mathbb{Q})$ is the direct image of the intersection complex of $M(n,0)$. Full support conjecture. The complex

$$R\chi(n,0)_*\operatorname{IC}(M(n,0),\mathbb{Q})$$

has full support on the whole basis of the Hitchin fibration. This is the main open conjecture after the effective decomposition theorem: it asks whether the degree-zero direct image has no summands supported on the complement of the reduced locus of the Hitchin base.