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.

 *(added later) Here's a counterexample. Choose a diffeomorphism $f$ between $\mathbb{C}$ and the disk $D$. Pulling back the Poincaré metric gives a Kähler metric such that $\dim H^1_{(2)}(\mathbb{P}-\{\infty\})=\infty$ because $f^*(z^ndz)$ gives an infinite family of harmonic $L^2$ forms. However intersection cohomology is finite dimensional.