Grothendieck's special case of the generalised Hodge conjecture

Let $Y$ be a smooth projective complex variety of pure dimension $n$ such that $h^{i,0}=0$ for some $i\leq n$. For a dense open subset $U\subseteq Y$, consider the restriction map

$$H^i_{\mathrm{Betti}}(Y,\mathbb{Q})\to H^i_{\mathrm{Betti}}(U,\mathbb{Q}).$$

Grothendieck's conjecture. There exists a dense open $U\subseteq Y$ such that this restriction map vanishes.

This is the stated special case of Grothendieck's generalised Hodge conjecture, concerning the coniveau of cohomology classes. The supplied text does not indicate whether this formulation has been resolved.