Totaro's birational product conjecture for anisotropic quadrics

Let $X$ be an anisotropic quadric over a field $k$. Write $\mathfrak{i}_1(X)$ for its first Witt index. A subquadric $X'\subset X$ is a quadric contained in $X$. Totaro's birational product conjecture. The quadric $X$ should be birational to

$$X'\times\mathbb{P}_k^{\mathfrak{i}_1(X)-1}$$

for some subquadric $X'\subset X$ of codimension $\mathfrak{i}_1(X)-1$. This is a more precise formulation of the ruledness conjecture; the source presents it as an open conjecture related to Totaro's work.