One of the results that can be deduced from the Artin Approximation Theorem (in its modern formulation) is the following: https://stacks.math.columbia.edu/tag/0CAV. It says that, if $S$ is a scheme such that $O_{S,s}$ is a G-ring, and $X,Y$ are schemes over $S$ locally of finite type, that satisfy: $$\widehat{O}_{X,x}\cong \widehat{O}_{Y,y} $$ for points $x,y$, then there exists a common étale neighborhood of $X,Y$ in the points $x,y$.
My questions are:
which counterexample do we have of this criterion? Are there noetherian schemes that have completed local rings isomorphic as above, but there exist no common étale neighborhood?
Are there at least examples of noetherian non G-rings? By now I have found just this one on wikipedia https://en.wikipedia.org/wiki/Excellent_ring#A_J-2_ring_that_is_not_a_G-ring.