  Faltings conjectured that under the p-adic Simpson correspondence, finite
dimensional p-adic representations of the geometric \'etale fundamental group
of a smooth proper p-adic curve X are equivalent to semi-stable Higgs bundles
of degree zero over X. In this article, we establish an equivalence between
these representations and Higgs bundles, whose underlying bundles potentially
admit a strongly semi-stable reduction of degree zero. We show that these Higgs
bundles are semi-stable of degree zero and investigate some evidence for
Faltings' conjecture.
