Bruzzo's conjecture on semistability and vanishing discriminant for Higgs bundles

Let $X$ be a smooth projective variety of dimension $n$, and let $\mathcal{E}=(E,\theta)$ be a Higgs bundle on $X$ of rank $r$. Define its discriminant by

$$\Delta(E)=2rc_2(E)-(r-1)c_1^2(E).$$

For a smooth projective curve $C$ and a morphism $f:C\longrightarrow X$, write $f^*\mathcal{E}$ for the induced Higgs bundle; say that $\mathcal{E}$ is curve Higgs semistable when $f^*\mathcal{E}$ is Higgs semistable for every such $f$. A polarization $H$ is an ample divisor class on $X$.

Bruzzo's conjecture. The following conditions are equivalent:

1. $\mathcal{E}$ is semistable with respect to some polarization $H$ and

$$\Delta(E)\cdot H^{n-2}=0.$$

2. For every morphism $f:C\longrightarrow X$ from a smooth projective curve $C$, the Higgs bundle $f^*\mathcal{E}$ is Higgs semistable; equivalently, $\mathcal{E}$ is curve Higgs semistable.

The implication from condition (1) to condition (2) is known, while the converse is the remaining direction of the conjecture.