The type II ring-isomorphism conjecture for locally measurable operator algebras

Let $M$ and $N$ be von Neumann algebras of type II. Let $LS(M)$ and $LS(N)$ denote their algebras of locally measurable operators. A ring isomorphism is a bijective ring homomorphism $\Psi\colon LS(M)\to LS(N)$. A real $^*$-isomorphism is a bijective real-linear $^*$-preserving algebra homomorphism. The type II ring-isomorphism conjecture. If $\Psi\colon LS(M)\to LS(N)$ is a ring isomorphism, then there exist an invertible operator $y\in LS(N)$ and a real $^*$-isomorphism $\psi\colon M\to N$ such that

$$\Psi(x)=y\psi(x)y^{-1}$$

for any $x\in LS(M)$. The conjecture extends the corresponding result for von Neumann algebras of type $\mathrm{I}_\infty$ or III. The source notes that little is known about the structure of $LS(M)$ for type II algebras and that even automatic real-linearity of such ring isomorphisms is unknown, including for approximately finite-dimensional type II$_1$ factors.