Cheeger–Goresky–MacPherson conjecture on intersection cohomology and $L^2$-cohomology

Let $X\subset \mathbb{P}^N$ be a projective variety, let $X_{\rm reg}$ be its regular locus, and let $ds^2_{\rm FS}$ be the Fubini–Study metric on $X_{\rm reg}$. Write $H^\ast_{(2)}(X_{\rm reg},ds^2_{\rm FS})$ for the corresponding $L^2$-cohomology and $IH^\ast(X,\mathbb{C})$ for the intersection cohomology of $X$. Cheeger–Goresky–MacPherson conjecture. There is an isomorphism

$$H^\ast_{(2)}(X_{\rm reg},ds^2_{\rm FS})\simeq IH^\ast(X,\mathbb{C}).$$

The conjecture proposes an analytic realization of intersection cohomology by square-integrable differential forms. Such an identification would provide analytic access to the Hard Lefschetz theorem and the pure Hodge structure on the intersection cohomology of a projective variety; the supplied text does not state its resolution.