Identification of intersection cohomology with L^2-cohomology for unipotent harmonic bundles

Let $S$ be a noncompact smooth curve with smooth compactification $\overline S$, let $j:S\hookrightarrow\overline S$ be the inclusion, and let $L_{\rho}$ be the local system associated with a unipotent harmonic bundle on $S$. Denote by $H^*_{\mathrm{int}}(\overline S,j_*L_{\rho})$ the relevant intersection cohomology and by $H^*_{(2)}(\overline S,L_{\rho})$ the $L^2$-cohomology computed using the Poincaré-like metric on $S$ and the harmonic metric on $L_{\rho}$. Identification conjecture. There exists a natural identification

H^*_{\mathrm{int}(\overline S,j_*L_{\rho})\cong H^*_{(2)}(\overline S,L_{\rho}).

For curves, this proposes the intersection-cohomological analogue of the preceding identification between $H^*(\overline S,j_*L_{\rho})$ and $L^2$-cohomology. The analogous statement in higher dimensions is known for variations of Hodge structure, but the general case remains open in the supplied context.