The semifinite embedding conjecture for nonassociative $cL^p$-spaces

Let $\mathcal{M}$ be a $\mathrm{JW}^*$-algebra equipped with a normal semifinite faithful trace $\tau$, and let $1{\leqslant} p<\infty$. Let $\mathrm{L}^p(\mathcal{M},\tau)$ denote the nonassociative $\mathrm{L}^p$-space associated with this data. Semifinite embedding conjecture. The space $\mathrm{L}^p(\mathcal{M},\tau)$ is isometric to a positively $1$-complemented subspace of a noncommutative $\mathrm{L}^p$-space associated with a semifinite von Neumann algebra.

This extends the proved finite-trace embedding theorem to normal semifinite faithful traces. The source states that particular cases are proved, while the general semifinite assertion remains open.