Chen–Ngô surjectivity conjecture for the Hitchin morphism

Let $X$ be a smooth proper scheme over an algebraically closed field of characteristic zero, and let $r\in\mathbb{N}$. Write $h^r_X:\mathscr{M}^r_X\to\mathscr{B}^r_X$ for the Hitchin morphism restricted to the spectral base.

Chen–Ngô's conjecture. For every point $s\in\mathscr{B}^r_X$, the fiber

$$(h^r_X)^{-1}(s)$$

is non-empty.

The conjecture asserts surjectivity of the spectral data morphism and asks whether every spectral datum is realized by a Higgs bundle. It is proved in the paper for $r$-small varieties, including varieties with numerically trivial canonical divisor, but is not established in general.