Bruzzo and Graña Otero conjecture for curve semistable Higgs bundles

Let $X$ be a smooth projective variety, let $r$ be the rank of a Higgs bundle, and let $\mathfrak{E}=(E,\varphi)$ be a curve semistable Higgs bundle over $X$. Its discriminant class is

$$\Delta(E)=\frac{1}{2r}c_2(\operatorname{End}(E))=c_2(E)-\frac{r-1}{2r}c_1(E)^2.$$

Bruzzo and Graña Otero conjecture. The Higgs bundle $\mathfrak{E}$ is semistable with respect to some polarization $H$ and $\Delta(E)=0$.

The conjecture asks whether curve semistability forces ordinary semistability for some polarization together with vanishing discriminant. It is known in several cases, including rank $2$, varieties with nef tangent bundle, various surfaces, simply connected Calabi–Yau varieties, and certain bundles with filtrations by H-nflat quotients of rank at most $2$; it remains open in general.