Chen–Ngô surjectivity of the Hitchin morphism onto the spectral base

Establish that for every spectral datum s in the spectral base B_X^r associated with a smooth proper scheme X over an algebraically closed field of characteristic 0 and a fixed rank r, the fiber (h_X^r)^{-1}(s) of the Hitchin morphism h_X^r: M_X^r → B_X^r from the moduli stack of rank r Higgs bundles on X to the spectral base is non-empty.

Background

For a smooth proper scheme X and rank r, the Hitchin morphism h_X^r maps the moduli stack of rank r Higgs bundles to the Hitchin base. Chen and Ngô introduced the spectral base B_X^r as a closed subscheme of the Hitchin base and showed that h_X^r factors through B_X^r, defining the spectral data morphism.

The conjecture asks for surjectivity of the spectral data morphism on k-points (equivalently, non-emptiness of every fiber), generalizing the well-known surjectivity in the curve case and extending confirmed cases such as certain surfaces and other special classes of varieties.