Chern-class vanishing conjecture for stable H-nflat Higgs bundles

Let $X$ be a smooth complex projective surface, and let $\mathfrak{E}=(E,\varphi)$ be a stable H-nflat Higgs bundle on $X$. Its Chern classes are the classes $c_i(E)$ in the Chow ring of $X$.

Chern-class vanishing conjecture. All Chern classes of $\mathfrak{E}$ vanish.

This conjecture is stated as equivalent to the Bruzzo–Graña Otero conjecture in the source, using the cited results. The surrounding discussion records several cases in which the equivalent conjecture is known, but the general assertion remains open.