The type II projection-lattice and locally measurable-ring conjecture

Let $M$ and $N$ be von Neumann algebras of type II. Write $\mathcal{P}(M)$ and $\mathcal{P}(N)$ for their projection lattices, and let $LS(M)$ and $LS(N)$ denote their algebras of locally measurable operators. A lattice isomorphism is an isomorphism of the projection lattices preserving their lattice structure; a ring isomorphism is a bijective ring homomorphism. The type II projection-lattice conjecture. If $\mathcal{P}(M)$ and $\mathcal{P}(N)$ are lattice isomorphic, or equivalently if $LS(M)$ and $LS(N)$ are ring isomorphic, then $M$ and $N$ are real $^*$-isomorphic, or equivalently $M$ and $N$ are Jordan $^*$-isomorphic. The source presents this as a weaker suspected statement because the stronger conjugacy description of ring isomorphisms is unknown for type II algebras. It also notes that the equivalence between projection-lattice isomorphism and ring isomorphism is being used in this setting, while the relevant structural questions for type II, especially type II$_1$, algebras remain open.