The axial central extension conjecture for Jordan algebras

Let $F$ be a field of characteristic different from $2$. Let $J$ be a Jordan algebra over $F$ generated by idempotents, and let a $\mathcal{J}(\frac{1}{2})$-axial central extension of $J$ be given with respect to any set of such generating idempotents. Axial central extension conjecture. Every such central extension is a Jordan algebra. The conjecture proposes that the Jordan identity is preserved by all $\mathcal{J}(\frac{1}{2})$-axial central extensions under these hypotheses; the preceding results establish it for several families, while the general case remains open.