  In SGA 2, Grothendieck conjectures that the \'etale fundamental group of the
punctured spectrum of a complete noetherian local domain of dimension at least
two with algebraically closed residue field is topologically finitely
generated. In this paper, we prove a weaker statement, namely that the maximal
pro-nilpotent quotient of the fundamental group is topologically finitely
generated. The proof uses $p$-adic nearby cycles and negative definiteness of
intersection pairings over resolutions of singularities as well as some
analysis of Lie algebras of certain algebraic group structures on deformation
cohomology.
