Colliot-Thélène–Sansuc–Swinnerton-Dyer conjecture for intersections of two quadrics

Let $k$ be a number field and let $X\subset \mathbb{P}^n_k$, with $n\geq 6$, be a non-conical geometrically integral intersection of two quadrics. Suppose that $X$ has a smooth $k_v$-point for every place $v$ of $k$. Colliot-Thélène–Sansuc–Swinnerton-Dyer conjecture. Then $X$ has a smooth $k$-point. This is a local-to-global principle for intersections of two quadrics, in the case considered by the paper; the supplied material does not state whether it has been resolved.