Weak Bloch conjecture for semistable Higgs bundles

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

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

where $\Delta(E)=2r c_2(E)-(r-1)c_1(E)^2$. Weak Higgs Bloch conjecture. For every $i\geq 1$,

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

where $B^i(X)$ is the group of codimension-$i$ cycles modulo algebraic equivalence. This is the weak, algebraic-equivalence version of the Higgs analogue of Bloch's conjecture and remains open in general.