Is every regular local ring a filtered colimit of essentially finitely generated regular local rings?

Question

Is every regular local ring $R$ a filtered colimit of regular local rings which are essentially of finite type over $\mathbb{Z}$ (i.e. localizations of finitely generated rings)?

For comparison, Popescu's theorem says that under the stronger assumption that $\mathrm{Spec}\,R\to\mathrm{Spec}\,\mathbb{Z}$ is a regular morphism, we have the stronger conclusion that $R$ is a filtered colimit of smooth $\mathbb{Z}$-algebras.

Answer 1

It seems it was proved recently by Popescu himself, see On a question of Swan (with an appendix by Kęstutis Česnavičius).