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Let X/s{X/s} be a proper log smooth scheme of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field κ of characteristic p>0{p>0}. In this article we prove that the cohomology of 𝒲(𝒪X){{\mathcal{W}}({\mathcal{O}}_{X})} is a finitely generated 𝒲(κ){{\mathcal{W}}(\kappa)}-module if the Yobuko height of …