Surjectivity conjecture for the Hitchin morphism onto the postulated image

Let $X$ be a proper smooth algebraic variety over an algebraically closed field $k$ of characteristic zero, let $\mathcal{M}_X$ be the moduli stack of Higgs bundles, and let $\mathcal{B}_X$ be the postulated image of the Hitchin morphism $h_X:\mathcal{M}_X\to\mathcal{A}_X$. Fiber non-emptiness conjecture. For every $b\in\mathcal{B}_X(k)$, the fiber

$$h_X^{-1}(b)$$

is non-empty. Thus every geometric point of the postulated image should be realized by a Higgs bundle; the source gives no resolution.