Invertible-summand conjecture for anisotropic quadrics

Let $Q$ be an anisotropic quadric over $k$, with defining quadratic form $q$, and let $\widetilde{M}_{\operatorname{K}(n)}(Q)$ be its $\operatorname{K}(n)$-kernel motive. Invertible-summand conjecture. If $\widetilde{M}_{\operatorname{K}(n)}(Q)$ contains an invertible direct summand, then

$$\operatorname{dim}_n q<2^n.$$

This is stated as the converse to a preceding proposition and follows from Kahn's descent conjecture; the latter is only known in special cases.