The generator problem: is every von Neumann algebra with separable predual singly generated?

Theorem 1.2: every type $\mathrm{II}_1$ factor with separable predual is $W^*(x)$ for one $x$ (two self-adjoint generators). Theorem 1.1: for an irreducible inclusion $P\subset M$ of $\mathrm{II}_1$ factors, $M$ separable, the unitaries $u$ with $W^*(P,u)=M$ are a dense $G_\delta$ in the 2-norm topology. With Willig's reduction this gives single generation of every von Neumann algebra with separable predual. Consequences: the generator invariants $G$ and $G_{sa}$ vanish, and Voiculescu's $\delta,\delta_0$ are not $W^*$-invariants on $L(\mathbb F_n)$, $n\ge3$. No explicit generator is constructed.

Theorem 1.2: every type $\mathrm{II}_1$ factor with separable predual is $W^*(x)$ for one $x$ (two self-adjoint generators). Theorem 1.1: for an irreducible inclusion $P\subset M$ of $\mathrm{II}_1$ factors, $M$ separable, the unitaries $u$ with $W^*(P,u)=M$ are a dense $G_\delta$ in the 2-norm topology. With Willig's reduction this gives single generation of every von Neumann algebra with separable predual. Consequences: the generator invariants $G$ and $G_{sa}$ vanish, and Voiculescu's $\delta,\delta_0$ are not $W^*$-invariants on $L(\mathbb F_n)$, $n\ge3$. No explicit generator is constructed.