In the article of Hesselholt [Hes05], a set of conjectures is laid out. Given a smooth scheme $X$ over the ring of integers $\mathcal{ O}_K$ of a $p$-adic field $K$, these conjectures concern the expected relation between log topological restriction homology $\mathrm{ TR}^r (X,M_X)$ and the absolute log de Rham--Witt complex $W_r\Omega_{ (X,M_X)}$. In this note, which is companion to [And24a], we discuss the case of a $p$-completely smooth $p$-adic formal scheme $X$ over $\mathrm{ spf} \mathcal{ O}_C$, where $C$ is the field of $p$-adic complex numbers. Along the way, we study the motivic filtration of log $\mathrm{ TR}^r$ and its $S^1$-homotopy fixed points, following ideas of [Bin+23].
Categories: math.AG math.KT
