Weak Kahn descent conjecture for quadratic forms

Let $q$ be an anisotropic quadratic form over $k$, let $K=k(q)$, and let $r$ be a quadratic form defined over $K$. Let $W_{\mathrm{nr}}(K/k)$ denote the unramified Witt ring. Weak Kahn descent conjecture. For some $n$, if

$$\operatorname{dim} r<2^n<\frac{1}{2}\operatorname{dim} q$$

and $r\in W_{\mathrm{nr}}(K/k)$, then $r$ is defined over $k$. The paper presents this as a weaker version of Kahn's conjecture and uses it in the study of invertible summands of Morava motives.