Weight-two injectivity conjecture for links of complex analytic spaces

Let $X$ be a complex analytic space that is $S_2$ and has pure dimension $\geq 3$, let $x\in X$ be a point, and let $\pi:\bar X\to X$ be the normalization with corresponding point $\bar x$. Weight-two injectivity conjecture. The pull-back map

$$\pi^*:H^2\bigl(\operatorname{link}(x,X),\mathbb C\bigr)\to H^2\bigl(\operatorname{link}(\bar x,\bar X),\mathbb C\bigr)$$

is injective on the weight-$2$ graded piece of the mixed Hodge structure. Such injectivity would imply most of the normalization conjecture for analytic spaces; the source does not state that it has been proved.