The coniveau conjecture for complete intersections

Let $X\subset\mathbb{P}^n$ be a smooth complete intersection of $r$ hypersurfaces of degrees $d_1\leq\cdots\leq d_r$, and let $H^{n-r}(X,\mathbb{Q})_{\mathrm{prim}}$ denote its primitive cohomology. A closed algebraic subset $Y\subset X$ has codimension $c$ when every irreducible component has codimension at least $c$. The coniveau conjecture for complete intersections. If

$$n\geq\sum_i d_i+(c-1)d_r,$$

then the restriction of $H^{n-r}(X,\mathbb{Q})_{\mathrm{prim}}$ to $X\setminus Y$ vanishes for some closed algebraic subset $Y\subset X$ of codimension $c$. This is stated as equivalent to the generalized Hodge conjecture in the complete-intersection setting and remains open in general.