Bayer-Fluckiger et al.'s conjecture on totally decomposable orthogonal involutions

Let $F$ be a field of characteristic not two, and let $(A,\sigma)$ be a central simple algebra with orthogonal involution of degree $2^n$ over $F$. An orthogonal involution is totally decomposable if it is a tensor product of involutions on quaternion algebras. For a field extension $K/F$, write $(A,\sigma)_K$ for the scalar extension, and call $(A,\sigma)$ hyperbolic when its scalar extension is hyperbolic. A field extension $K/F$ is a splitting field of $A$ when $A_K$ is split, and an involution on a split algebra is adjoint to a Pfister form when it is the adjoint involution of such a form.

Bayer-Fluckiger et al.'s conjecture. The following statements are equivalent:

1. $(A,\sigma)$ is totally decomposable.
2. For every field extension $K/F$, $(A,\sigma)_K$ is either anisotropic or hyperbolic.
3. For every splitting field $K$ of $A$, $(A,\sigma)_K$ is adjoint to a Pfister form.

The conjecture concerns characterizations of totally decomposable orthogonal involutions through their behavior after field extensions and over splitting fields. The source states that its characteristic-two analogue is settled in the paper, but gives no resolution here for the conjecture in characteristic different from two.