Twistor space for local systems on an open curve

Let $X=\overline{X}-D$ be a smooth quasi-projective curve. In arXiv:2110.12300 we constructed a Deligne-Hitchin modui space with Hecke gauge groupoid for connections of rank $2$. We extend this construction to the case of any rank $r$, although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.