Shapiro's conjecture on split decompositions of algebras with involution

Let $(A,\sigma)$ be a product of $r$ quaternions with involution. Suppose that $A$ is split.

Shapiro's conjecture. $(A,\sigma)$ admits a decomposition as a tensor product of $r$ quaternion algebras with involution in which each quaternion algebra is split.

This conjecture concerns whether a product of quaternions with involution whose underlying algebra is split can always be represented using split quaternion factors. The source gives no resolution, so the conjecture remains open.