The converse to Karpenko's non-ruledness theorem for anisotropic quadratic forms

Let $q$ be an anisotropic quadratic form over a field $k$. Its first Witt index is the Witt index of $q$ after extending scalars to the function field of the projective quadric defined by $q$, and $q$ is ruled over $k$ when its projective quadric is ruled.

Converse to Karpenko's theorem. If the first Witt index of $q$ is greater than $1$, then $q$ is ruled over $k$.

Karpenko's theorem says that an anisotropic quadratic form with first Witt index $1$ is not ruled. This conjecture proposes the converse and is supported in the source by a proof for quadratic forms of dimension at most $9$.