Conjecture on complete boundedness of the Jordan component in Yeadon factorizations

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 an isometry with a Yeadon factorization $T=wbJ$, where $J$ is a normal $*$-Jordan homomorphism. Yeadon-factorization conjecture. If $T$ is completely bounded, then $J$ is completely bounded. This is identified in the text as the remaining reduction for the reverse direction of the preceding conjecture; no resolution is supplied in the given material.