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Let k be a perfect field of characteristic p>0 and let X be a proper scheme over W(k) with semistable reduction. I shall define a log-Chow group for the special fibre X_k and give an interpretation in terms of logarithmic Milnor K-theory. Then, by gluing a logarithmic variant of the Suslin-Voevodsky motivic complex to a log-syntomic complex along the logarithmic Hyodo-Kato Hodge-Witt sheaf, I will prove that an element of the r-th log-Chow group of X_k formally lifts to the continuous log-Chow group of X if and only if it is “Hodge” (i.e. its log-crystalline Chern class lands in the r-th step of the Hodge filtration of the generic fibre of X under the Hyodo-Kato isomorphism). This simultaneously generalises a result of Yamashita (which is the case r=1), and of Bloch-Esnault-Kerz (which is the case of good reduction). This is joint work with Andreas Langer.