Chen–Ngô's spectral-base image conjecture for the Hitchin morphism

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, let $X$ be an irreducible smooth projective variety over $\mathbb{K}$, and let $G$ be a reductive group over $\mathbb{K}$ of rank $n$. A $G$-Higgs bundle is a pair $(E,\theta)$ consisting of a principal $G$-bundle $E$ and an integrable section

$$\theta\in H^0(X,\operatorname{ad}(E)\otimes_{\mathscr{O}_X}\Omega_X^1),\qquad \theta\wedge\theta=0.$$

Let $\mathscr{M}_{X,G}$ be the moduli stack of $G$-Higgs bundles, let $\mathscr{A}_{X,G}$ be the Hitchin base, and let $h_{X,G}:\mathscr{M}_{X,G}\rightarrow\mathscr{A}_{X,G}$ be the Hitchin morphism. Let $\mathscr{B}_{X,G}\subseteq\mathscr{A}_{X,G}$ be the spectral base introduced by Chen and Ngô, and let $\operatorname{sd}_{X,G}:\mathscr{M}_{X,G}\rightarrow\mathscr{B}_{X,G}$ be the spectral morphism. Chen–Ngô's spectral-base image conjecture. For every $b\in\mathscr{B}_{X,G}(\mathbb{K})$, the fibre $h_{X,G}^{-1}(b)$ is nonempty; equivalently, the image of $h_{X,G}$ is exactly the spectral base $\mathscr{B}_{X,G}$. The conjecture remains open even for $G={\rm GL}_n$; known cases include $\dim X=2$ with $G={\rm GL}_n$ arbitrary and arbitrary $\dim X$ with $G={\rm GL}_2$.