Isometric decomposition conjecture for von Neumann factors

Let $\mathfrak{R}$ be a type-$\mathrm{II}_{\infty}$ or type-$\mathrm{III}$ factor acting on a separable Hilbert space, and let $W_e^{\mathfrak{R}}(A)$ denote its essential numerical range. An isometric decomposition is a sequence $\{V_i\}_{i=1}^{\infty}$ of isometries in $\mathfrak{R}$ satisfying

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

Let $\mathcal{D}$ be the specified set occurring in the source. Isometric decomposition 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

$$V_i^*AV_i=X_i\qquad\text{for all }i.$$

The surrounding section presents this as a possible extension of operator-pinning results from $\mathrm{L}(\mathcal{H})$ to type-$\mathrm{II}_{\infty}$ and type-$\mathrm{III}$ factors; the supplied text gives no resolution.