I'm not an expert, but you don't seem to be getting any answers.
First of all, I would be surprised if holds for any Kähler metric (e.g. for one with really bad singularities along $D$) but I don't have a counterexample. Regarding the case of Fubini-Study metric, for isolated singularities, I believe it was settled positively by Ohsawa: "Cheeger-Goreski [Goresky]-MacPherson's conjecture for the varieties with isolated singularities." Math. Z. 206 (1991). I don't know the status in general. I believe that there was a gap in a later paper by author on this.
There has been a lot of work with other metrics. For example, you can look at work on Zucker's conjecture, which was proved by Looijenga and Saper-Stern. Also for Poincaré type metrics, allowing coefficients in a variation of Hodge structure, there is the work by Cattani-Kaplan-Schmid and Kashiwara-Kawai. This provides the analytic basis for Saito's theory of Hodge modules.