Specialization for the pro-\'etale fundamental group

For a formal scheme $\mathfrak{X}$ of finite type over a complete rank one valuation ring, we construct a specialization morphism $$\pi^{\rm dJ}_1(\mathfrak{X}_\eta) \to \pi^{\rm proet}_1(\mathfrak{X}_k)$$ from the de Jong fundamental group of the rigid generic fiber to the Bhatt-Scholze pro-\'etale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of $\mathfrak{X}$ and normalizations of the irreducible components of $\mathfrak{X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy \'etale descent.