H. A. Dye showed that a type II or III factor $R$ is determined, up to *-algebraic isomorphism or anti-isomorphism, by the ortholattice-isomorphism type of its projection lattice ("ortholattice-isomorphism" meaning bijection between projection lattices that preserves order and orthogonality), but has anyone figured out how much we can determine about $R$ *just from the order-isomorphism type* of its projection lattice?

Dye says as of 1955 it was open whether "order-isomorphic" is equivalent to "ortholattice-isomorphic" in this context. Has progress been made on this question? I've been told that the Levy collapse poset* embeds densely into the projection lattice of any non-commutative non-type-I factor with separable predual, from which I think one could show that the lattice's order type tells you nothing more about such a factor. But I don't know if this was meant in the sense of "this is an established fact" or "this seems very likely."

[I asked this recently on math.stackexchange but it's likely too specialized ... any help hugely appreciated!]

[Also, I initially mentioned a paper by C. Heunen and M. Reyes, but as @BertLindenhovius explains below, their example gave only a pair of order-isomorphic projection lattices belonging to algebras that are not *-isomorphic; to be relevant to my question, the algebras would need to be neither *-isomorphic nor *-anti-isomorphic.]

*more precisely the Levy collapse of $2^\omega$ to $\omega$, i.e. the set of finite sequences of ordinals $< 2^\omega$, ordered by reverse inclusion.