Künneth conjecture for $p$-power torsion Azumaya algebras

Let $X$ be a quasi-compact and quasi-separated scheme in which $p$ is invertible, and let $A$ and $A'$ be $p$-power torsion Azumaya algebras over $X$. The Künneth map is

$$K(A)\otimes_{K(X)}K(A')\to K(A\otimes_X A').$$

Here $K(-)$ denotes algebraic $K$-theory, and a map is a $K(1)$-local equivalence when it becomes an equivalence after $K(1)$-localization. Künneth conjecture. The displayed Künneth map is a $K(1)$-local equivalence. This is not a consequence of the preceding result because not all $p$-power torsion Azumaya algebras are split by $p$-power Galois covers, even in the affine case. The source gives no resolution.