Chow-theoretic Chern class conjecture for stable Higgs bundles

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$, let $H$ be an ample divisor on $X$, and let $(E,\theta)$ be a slope $H$-stable Higgs vector bundle of rank $r$ satisfying

$$\int_X\Delta(E)\cdot H^{d-2}=0,$$

where $\Delta(E)=2r c_2(E)-(r-1)c_1(E)^2$. Assume that for every smooth projective variety $Y/k$ and every generically finite map $f:Y\to X$, the pullback Higgs sheaf $f^*(E,\theta)$ is slope stable with respect to some ample divisor. Stable Higgs Chern class conjecture. For every $i\geq 1$,

$$r^i c_i(E)=\binom{r}{i}c_1(E)^i\quad\text{in }\operatorname{CH}^i(X)_{\mathbb Q}.$$

The numerical discriminant condition gives the same identities in de Rham cohomology, and the conjecture asks for their Chow-theoretic refinement. It is open in general.