Grothendieck's finite generation conjecture for punctured-spectrum fundamental groups

Let $A$ be a complete noetherian local ring with algebraically closed residue field $F$ and maximal ideal $\mathfrak{m}$. Let $p$ be the characteristic of $F$ if it is positive and let $p = 1$ otherwise. Assume that the irreducible components of $\operatorname{Spec} A$ have dimension at least $2$, and that the scheme $\operatorname{Spec} A \setminus \{\mathfrak{m}\}$ is connected.

Grothendieck's conjecture. The following statements hold:

1. The étale fundamental group $\pi_{1}(\operatorname{Spec} A \setminus \{\mathfrak{m}\})$ is topologically finitely generated.
2. The maximal pro-prime-to-$p$ quotient of $\pi_{1}(\operatorname{Spec} A \setminus \{\mathfrak{m}\})$ is topologically finitely presented.

The conjecture, made in SGA 2, concerns the finiteness properties of the étale fundamental group of a punctured spectrum. The paper proves the weaker assertion that the maximal pro-nilpotent quotient is topologically finitely generated, so the two stated finiteness claims remain unresolved here.