Strong semistability of Higgs bundles over curves

In this paper we complete the study of the Lan-Sheng-Zuo conjecture proposed in arXiv:1210.8280 for the curve case. Precisely, we prove that every semistable Higgs bundle is strongly semistable for curves of genus $g\le 1$, and over any curves of genus $g\ge 2$ construct explicit examples of semistable Higgs bundles of arbitrary big rank (the first example is $p=2,r=3$) which are not strongly semistable. These results are complementary to the strongly semistability theorem of Lan-Sheng-Yang-Zuo and Langer for semistable Higgs bundles of small rank.