The Higgs-bundle curve-semistability characterization

Let $(Y,H)$ be a complex polarized variety, and let $(F,\phi)$ be a Higgs bundle on $Y$. For a pair $(C,f)$, where $C$ is an irreducible, nonsingular, projective curve and $f:C\to Y$ is a morphism, write $f^*(F,\phi)$ for the pull-back Higgs sheaf; say that $(F,\phi)$ is curve-semistable if $f^*(F,\phi)$ is semistable for every such pair.

Curve-semistability characterization. The following are equivalent:

$$\begin{array}{ll} \text{(i)} & (F,\phi)\text{ is semistable with vanishing discriminant in }H^4(Y,\mathbb Q);\\ \text{(ii)} & (F,\phi)\text{ is curve-semistable.} \end{array}$$

This is the Higgs-sheaf analogue of the characterization of semistable locally free sheaves with vanishing discriminant by curve-semistability. The implication from (i) to (ii) was proved, while the converse is the remaining direction of the stated equivalence.