  Let $K$ be a field of characteristic zero complete for a discrete valuation,
with perfect residue field of characteristic $p>0$, and let $K^+$ be the
valuation ring of $K$. We relate the log-crystalline cohomology of the special
fibre of certain affine $K^+$-schemes $X=\text{Spec}(R)$ with semi-stable
reduction to the Galois cohomology of the fundamental group of the geometric
generic fibre $\pi_1(X_{\bar{K}})$ with coefficients in a Fontaine ring
constructed from $R$. This is based on Faltings' theory of almost \'etale
extensions.
