Totaro's ruledness conjecture for anisotropic quadrics

Let $X$ be an anisotropic quadric over a field $k$. Write $\mathfrak{i}_1(X)$ for its first Witt index, and say that $X$ is ruled if it is birational to $Y\times\mathbb{P}_k^1$ for some variety $Y$ over $k$. Totaro's ruledness conjecture. The quadric $X$ is ruled if and only if

$$\mathfrak{i}_1(X)>1.$$

The theorem that anisotropic quadrics with first Witt index $1$ are not ruled proves one direction; the converse and the full equivalence remain open in the source.