The converse to Karpenko's theorem on ruled quadrics

Let $q$ be an anisotropic quadratic form over a field $k$. Its first Witt index is the Witt index of $q$ over the function field of its associated projective quadric. Converse to Karpenko's theorem. If the first Witt index of $q$ is greater than $1$, then the associated projective quadric is ruled over $k$. Karpenko's theorem states that anisotropic quadratic forms with first Witt index equal to $1$ have non-ruled associated projective quadrics. The converse is conjectured in any characteristic; the source notes that it is proved there for quasilinear forms of every dimension.