Isomorphism of complete local noetherian rings and strong approximation

About a year ago Angus Macintyre raised the following question. Let $A$ and $B$ be complete local noetherian rings with maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$ such that $A/\mathfrak{m}^n$ is isomorphic to $B/\mathfrak{n}^n$ for every $n$. Does it follow that $A$ and $B$ are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.