The optimistic conjecture on canonical Banach topologies for cohomology

Let $F$ be the coefficient field, let $\mathcal{R}_F$ be its Robba ring, and let $\mathcal{D}$ be a $(\varphi,\Gamma)$-module over $\mathcal{R}_F$. For a representation $V$ of $\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathcal{K}_\infty)$, write $\mathbf{D}^\dag_{\mathrm{rig}}(V)$ for its associated $(\varphi,\Gamma)$-module and $H^1(\mathcal{K}_\infty,V)$ for its cohomology. Optimistic conjecture. The group $H^1(\mathcal{D})$ carries a natural Banach-space topology, functorial in $\mathcal{D}$, which agrees with the Banach-space structure on $H^1(\mathcal{K}_\infty,V)$ when $\mathcal{D}=\mathbf{D}^\dag_{\mathrm{rig}}(V)$. The conjecture would provide a canonical topological framework for the cohomology of $(\varphi,\Gamma)$-modules. The source states that it has not been proved in full generality.