As a further addendum, there are different "aspects" in which you could want the bound to be subconvex. As a result, even for a given family of L-functions) there is more than one "subconvexity problem." In this case, Nelson's bound is not subconvex in the non-Archimedean aspect. This is another reason why it doesn't seem quite right to me to say without qualification that "the subconvexity problem has been solved."