Kahn's low-dimensional unramified Witt-ring conjecture

Let $F$ be a field of characteristic different from $2$, let $\varphi$ be an anisotropic quadratic form over $F$, and let $F(\varphi)$ be the function field of its projective quadric. Let $W_{\operatorname{nr}}(F(\varphi))$ be the unramified Witt ring, and let $[\tau]$ be the Witt class of a quadratic form in this ring. Kahn's conjecture. If

$$\operatorname{dim}(\tau)<\frac{1}{2}\operatorname{dim}(\varphi),$$

then $\tau$ is defined over $F$: there exists a quadratic form $\sigma$ over $F$ such that

$$\tau\simeq\sigma_{F(\varphi)}.$$

The conjecture asserts that the low-dimensional part of the unramified Witt ring of a quadric comes from the base field. The source uses it as a stronger conjectural input toward the refined splitting-pattern conjecture; no resolution is given there.