The non-abelian Hodge correspondence for extensions on quasi-projective curves

Let $\bar{X}$ be a smooth projective connected complex curve, $D\subset \bar{X}$ be a finite set of reduced points, and $X=\bar{X}\setminus D$. We give a moduli-theoretic construction of an exact equivalence between logarithmic flat bundles on $(\bar{X},D)$ with nilpotent residues and semistable logarithmic Higgs bundles of degree zero with nilpotent residues. This extends the tame Simpson-Mochizuki correspondence on polystable bundles.