On the nilpotent residue non-abelian Hodge correspondence for higher-dimensional quasiprojective varieties

In arXiv:2408.16441, the authors proved that on a projective log smooth variety $(\bar{X},D)$ there is a continuous bijection between the moduli space $M_{\mathrm{D}\mathrm{o}\mathrm{l}}^{\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{p}}(\bar{X},D)$ of logarithmic Higgs bundles with nilpotent residues and the moduli space $M_{\mathrm{D}\mathrm{R}}^{\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{p}}(\bar{X},D)$ of logarithmic connections with nilpotent residues. In this note, we argue that the map is a homeomorphism.