Trace-class approximation by direct sums of irreducible operators for type II_1 factors

Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{B}(\mathcal{H})$ denote the bounded operators on $\mathcal{H}$. For an operator $T\in\mathcal{B}(\mathcal{H})$, write $W^*(T)$ for the von Neumann algebra generated by $T$, and let $\|K\|_1$ denote the trace-class norm.

Type II$_1$ factor approximation conjecture. Suppose that $T$ is an operator in $\mathcal{B}(\mathcal{H})$ such that $W^*(T)$ is a type $\mathrm{II}_1$ factor. Then for every $\varepsilon>0$, there exists a trace-class operator $K$ in $\mathcal{B}(\mathcal{H})$ with $\|K\|_1<\varepsilon$ such that $T+K$ is a direct sum of at most countably many irreducible operators.

This conjecture concerns density in the trace-class norm of operators that decompose as countable direct sums of irreducible operators, for operators generating type $\mathrm{II}_1$ factors. The paper proposes it in the context of single generators of type $\mathrm{II}_1$ factors; its resolution is not supplied in the given text.