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

Let $Y\subset \mathbb{C}\mathbb{P}^N$ be a projective variety. Write $H^k_{(2)}(Y-\operatorname{Sing}Y)$ for the $L^2$-de Rham cohomology of its regular part, with respect to the restricted Fubini–Study metric, and let $IH^k(Y)$ denote its middle-perversity intersection cohomology.

Cheeger–Goresky–MacPherson conjecture. There is a natural isomorphism

$$H^k_{(2)}(Y-\operatorname{Sing}Y)\cong IH^k(Y).$$

The conjecture connects analytic $L^2$-cohomology on singular varieties with intersection cohomology. It is proved when $Y$ has only isolated singularities, but remains open in general.