Conjecture on completely bounded isometries of noncommutative Lp-spaces

Let $M$ be a finite von Neumann algebra and let $N$ be a semifinite von Neumann algebra. Suppose $1{\leqslant} p<\infty$ with $p\ne 2$, and let

$$T\colon \mathrm{L}^p(M)\to \mathrm{L}^p(N)$$

be a not necessarily surjective isometry. Conjecture on completely bounded isometries. The map $T$ is completely bounded if and only if there exist a decomposition $M=M_1\oplus M_2$, with $M_1$ and $M_2$ von Neumann algebras, and an integer $n{\geqslant} 1$ such that the restriction

$$T|\mathrm{L}^p(M_1)\colon \mathrm{L}^p(M_1)\to \mathrm{L}^p(N)$$

is a complete isometry and $M_2$ is $n$-minimal. The forward implication is presented as the conjectural part, while the reverse implication is established in the surrounding discussion; the claim concerns the structure of completely bounded isometries on noncommutative $\mathrm{L}^p$-spaces for finite von Neumann algebras.