TR with logarithmic poles and the de Rham-Witt complex

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 ${\mathrm{T}\mathrm{R}}^r(X,M_X)$ and the absolute log de Rham–Witt complex $W_r{\mathrm{\Omega }}_{(X,M_X)}$. In this note, which is companion to [And24a], we discuss the case of a $p$-completely smooth $p$-adic formal scheme $X$ over $\mathrm{s}\mathrm{p}\mathrm{f}{\mathcal{O}}_C$, where $C$ is the field of $p$-adic complex numbers. Along the way, we study the motivic filtration of log ${\mathrm{T}\mathrm{R}}^r$ and its $S^1$-homotopy fixed points, following ideas of [Bin+23].