Strong uncountable cofinality for unitary groups of von Neumann algebras

We show that unitary groups of II${}_1$ factors and of properly infinite von Neumann algebras have strong uncountable cofinality. In particular, we obtain a short alternative proof for the strong uncountable cofinality of $U({\mathrm{\ell }}^2(\mathbb{N}))$, which was first proven by Ricard and Rosendal.