Characteristic class version of the Hodge index theorem

Let $X$ be a compact pure-dimensional complex algebraic variety. The Goresky–MacPherson homology $L$-class of $X$ is denoted by $L_*(X)$, and its intersection cohomology Hirzebruch class by $IT_{y*}(X)$. Characteristic class version of the Hodge index theorem. One has

$$L_*(X)=IT_{1*}(X)$$

in $H_{2*}(X;\mathbf{Q})$. This conjecture lifts the singular Hodge index equality $\sigma(X)=I\chi_1(X)$ from degrees to characteristic classes. It is known for rational homology manifolds, and in particular in the projective case; the general case for compact pure-dimensional complex algebraic varieties remains open.