Halmos' reducible-operator conjecture for separable factors

Let $\mathcal{M}$ be a separable factor. An operator $T\in\mathcal{M}$ is reducible if there is a projection $P\in\mathcal{M}$ with $P\ne 0,I$ and $PT=TP$. Halmos' conjecture. The set of reducible operators is nowhere norm-dense in $\mathcal{M}$ if and only if $\mathcal{M}$ is a finite factor. The claim extends the known finite-dimensional and non-$\Gamma$ type $\mathrm{II}_1$ results and contrasts them with norm-density of reducible operators in separable properly infinite factors; the remaining cases concern finite factors not covered by those results.