The $A_{inf}$-cohomology in the semistable case

For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable ${\mathcal{O}}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same ${\mathcal{O}}_K$-lattice inside $H_{dR}^i(X\mathrm{/}K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt–Morrow–Scholze on the construction and the analysis of an $A_{inf}$-valued cohomology theory of $p$-adic formal, proper, smooth ${\mathcal{O}}_{\bar{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{inf}$-cohomology to the $p$-adic '{e}tale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.