An integral Hyodo--Kato isomorphism

Let $\mathscr{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$, and let $\mathfrak{X}$ be a proper formal scheme over $\mathscr{O}_K$ with semistable reduction. Answering a question of Fontaine and Jannsen, a classical theorem of Hyodo and Kato gives a rational identification between the log crystalline cohomology of the special fiber $\mathfrak{X}_0$ over $W(k)^0$ with the de Rham cohomology of the generic fiber $\mathfrak{X}_K$. In this note, we use a log variant of the saturated de Rham--Witt complex of Bhatt--Lurie--Mathew to prove that such a comparison holds integrally.