The factor-valued pinching conjecture for type-II-infinity and type-III factors

Let ${\mathfrak{R}}$ be a type-${\mathrm{II}}_{\infty}$ or type-${\mathrm{III}}$ factor, let $W_e^{\mathfrak{R}}(A)$ denote its essential numerical range, and let ${\mathcal{D}}$ be the set used in the preceding pinching results. An isometric decomposition of ${\mathfrak{R}}$ is a sequence $\{V_i\}_{i=1}^{\infty}$ of isometries in ${\mathfrak{R}}$ such that

$$\sum_{i=1}^{\infty}V_iV_i^*=I.$$

Factor-valued pinching conjecture. If $A\in{\mathfrak{R}}$ satisfies $W_e^{\mathfrak{R}}(A)\supset{\mathcal{D}}$ and $\{X_i\}_{i=1}^{\infty}\subset{\mathfrak{R}}$ satisfies $\sup_i\|X_i\|<1$, then there exists an isometric decomposition $\{V_i\}_{i=1}^{\infty}$ of ${\mathfrak{R}}$ such that

$$V_i^*AX_i=X_i$$

for all $i$. The conjecture proposes that the pinching theorem extends from ${\mathrm{L}}({\mathcal{H}})$ to these von Neumann factors, giving affirmative answers to the preceding extension questions; the source indicates that this is suggested by related decompositions of positive operators, but does not establish the factor version.