The weak weak approximation conjecture for smooth unirational varieties

Let $K$ be a number field. A $K$-variety $X$ satisfies weak weak approximation (WWA) if there is a finite set $S_0$ of places of $K$ such that, for every finite set $S_1$ of places disjoint from $S_0$, the diagonal image of $X(K)$ is dense in $\prod_{v\in S_1}X(K_v)$. A variety $X$ has the Hilbert property (HP) when it admits a Hilbertian set of rational points in the sense used in the paper. Weak weak approximation conjecture. Every smooth unirational variety over a number field satisfies WWA, and consequently has HP. This conjecture predicts that the approximation properties of affine space extend to all smooth unirational varieties.