The semistable comparison conjecture for smooth affinoid rigid spaces

Let $C$ be the completed algebraic closure of a $p$-adic field, and let $X$ be a smooth affinoid rigid space over $C$. Write $\boldsymbol{\Omega}^j(X)$ for the space of $j$-forms, $d$ for the de Rham differential, $H^i_{\operatorname{pro\acute{e}t}}(X,\mathbb{Q}_p(i))$ for rational pro-étale cohomology, $H^i_{\operatorname{HK}}(X)$ for Hyodo–Kato cohomology, $B_{\log}$ for the logarithmic period ring, and $N$ and $\varphi$ for its monodromy and Frobenius operators. The semistable comparison conjecture. For every $i\geq 0$, there is a short exact sequence in $\operatorname{Mod}_{\mathbb{Q}_p}^{\operatorname{solid}}$

$$0\to \Omega^{i-1}(X)/\ker d \to H^i_{\operatorname{pro\acute{e}t}}(X,\mathbb{Q}_p(i))\to (H^i_{\operatorname{HK}}(X)\otimes^{\blacksquare}_{\breve F}B_{\log})^{N=0,\varphi=p^i}\to 0.$$

This is the affinoid analogue of the corresponding theorem for smooth Stein spaces and would give a refined description of rational pro-étale cohomology in terms of de Rham and Hyodo–Kato data. The claim is presented as a conjecture in the source, and no resolution is supplied.