Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically preserves $0$, $1$, order, and commutativity) between two complex von Neumann algebras $M$ and $N$, and $M$ has no type $I_2$ direct summand, then $\theta$ extends to a Jordan isomorphism between $M$ and $N$.

I am interested whether the same theorem still holds for real von Neumann algebras. The answer to the question Type III factor representation by https://mathoverflow.net/users/46855/user46855 contains the claim that it holds for real $AW$*-algebras with no abelian or type $I_2$ direct summand, together with a sketch of a proof. Hence it seems to be true if we also exclude the abelian summand. But I cannot find a reference to a complete proof of this result.

So my question is as follows. Does anyone have a reference to the following result: if $\theta$ is a projection orthoisomorphism between real von Neumann algebras $M$ and $N$, with $M$ having no abelian or type $I_2$ direct summand, does $\theta$ extend to a Jordan isomorphism between $M$ and $N$?