Semistable comparison conjecture for torsion in p-adic étale cohomology

Let $\mathfrak{X}$ be a semistable formal scheme over $\mathcal{O}_C$, meaning that it has étale-locally semistable coordinates, and equip it with its canonical log structure. Let $k^0$ denote $k$ with the log structure associated with $\mathbb{Z}_{\ge 0}\to k$, sending $1$ to $0$ and $0$ to $1$. Suppose that $\mathfrak{X}$ admits an open affine covering $\{\mathfrak{U}_r\}_{r\in \mathbb{Z}_{\ge 0}}$ with $\mathfrak{U}_r\subseteq \mathfrak{U}_{r+1}$.

Semistable comparison conjecture. For all $i\in \mathbb{Z}$ and $n\in \mathbb{Z}_{\ge 0}$, there is a natural isomorphism

$$H^{i+1}_{\operatorname{pro\acute{e}t}}(\mathfrak{X}_{C}, \mathbb{Z}_p)[p^n]\cong H^1(\mathfrak{X}_k/k^0, W\Omega_{\log}^i)[p^n].$$

Here $W\Omega_{\log}^i$ denotes logarithmic de Rham–Witt cohomology for log schemes.

This conjecture proposes extending the comparison theorem for formal schemes with smooth reduction to the semistable reduction case. The logarithmic de Rham–Witt theory used on the right-hand side is attributed to Lorenzon; the source provides no resolution status.