$L^2$-cohomology of a variation of Hodge structure for an infinite covering of an open curve ramified at infinity

Let $X$ be a compact Riemann surface, $\mathrm{\Sigma }$ a finite set of points and $M=X\setminus \mathrm{\Sigma }$. We study the $L^2$ cohomology of a polarized complex variation of Hodge structure on a Galois covering of the Riemann surface of finite type $M$. In this article we treat the case when the covering comes from a branched covering of $X$, and where $M$ is endowed with a metric asymptotic to a Poincaré metric. We prove that after tensorisation with the algebra of affiliated operators, the $L^2$ cohomology admits a pure Hodge structure.