McDuff proved that there exist continuum many non-isomorphic (separable) II${}_1$ factors. I would like to politely ask whether it is known/open if one can find $2^{\mathfrak{c}}$ (or at least $\mathfrak{c}^+$) many such factors.

My feeling is that this is not possible to construct more than $\mathfrak{c}$ separable von Neumann algebras by a simple cardinality argument. The ball of $B(H)$ for $H$ separable, is metrisable under the ultraweak topology, so it has at most $\mathfrak{c}$ ultraweakly closed subsets. So we cannot have more than $\mathfrak{c}$ different balls, and consequently, have more than $\mathfrak{c}$ non-isomorphic algebras. Is this correct?