The abelian-root-group conjecture for proper Moufang sets

Let $(X,(U_x)_{x\in X})$ be a proper Moufang set, meaning that its little projective group is not sharply $2$-transitive, and suppose that each root group $U_x$ is abelian. A quadratic Jordan division algebra is a quadratic Jordan algebra in which every nonzero element is invertible; for such an algebra $J$, write $\mathbb{M}(J)$ for the Moufang set associated with $J$. Abelian-root-group conjecture. There is a field $k$ and a quadratic Jordan division algebra $J$ over $k$ such that

$$(X,(U_x)_{x\in X})$$

is isomorphic to $\mathbb{M}(J)$. There has been progress in proving this conjecture, but in general it remains open; it would classify proper Moufang sets with abelian root groups via quadratic Jordan division algebras.