I'm curious if anyone knows a reference for the following. It seems like someone must have done this somewhere, but I couldn't find a reference.

Recall that an excellent reduced noetherian ring $R$ is called weakly normal if any finite birational map $R \subset S$ such that

1. The induced map on spectra is a bijection.
2. Every induced residue field extension $R_q/q \subset S_p/p{ }$ is purely inseparable.

is an isomorphism.

(In other words, there can be no such proper, finite birational extension of $R$ satisfying conditions (i) and (ii) ).

Suppose now that $R$ is weakly normal and local with maximal ideal $\mathfrak{m}$.

Question Is the completion of $R$ at the maximal ideal also weakly normal?

For the case of seminormality (replace inseparable in (ii) with isomorphism), this is known, see Greco-Traverso. It may be that one can mimic that argument (although there are a couple possibly small issues that I see).

I'm curious if anyone knows a reference for the following. It seems like someone must have done this somewhere, but I couldn't find a reference.

Recall that an excellent reduced noetherian ring $R$ is called weakly normal if any finite birational map $R \subset S$ with $S$ also reduced such that

1. The induced map on spectra is a bijection.
2. Every induced residue field extension $R_q/q \subset S_p/p{ }$ is purely inseparable.

is an isomorphism.

(In other words, there can be no such proper, finite birational extension of $R$ satisfying conditions (i) and (ii) ).

Suppose now that $R$ is weakly normal and local with maximal ideal $\mathfrak{m}$.

Question Is the completion of $R$ at the maximal ideal also weakly normal?

For the case of seminormality (replace inseparable in (ii) with isomorphism), this is known, see Greco-Traverso. It may be that one can mimic that argument (although there are a couple possibly small issues that I see).