Banach cohomology and comparison conjecture for $p$-adic period domains

Let $I \triangleleft e \triangle$ and let $i \in \mathbb{N}_0$. The cohomology groups

$$H^i(\mathscr{F}^{\operatorname{rig}} \backslash Y_I^-(\epsilon),\mathcal{E}_\lambda),\qquad H^i_{Y_I^-(\epsilon)}(\mathscr{F}^{\operatorname{rig}},\mathcal{E}_\lambda)$$

are $K$-Banach spaces, with the algebraic cohomology groups $H^i(\mathscr{F} \backslash Y_I,\mathcal{E}_\lambda)$ and $H^i_{Y_I}(\mathscr{F},\mathcal{E}_\lambda)$, respectively, as dense subspaces. Here $\mathscr{F}$ is the relevant flag variety, $Y_I$ is the corresponding closed subset, $Y_I^-(\epsilon)$ is its rigid-analytic neighbourhood, $\mathcal{E}_\lambda$ is the coefficient line bundle, and $\epsilon_m$ is the specified cofinal sequence of radii. Write $\hat{H}^i_{Y_I,m}$ for the corresponding completed cohomology group.

Banach cohomology and comparison conjecture. There is an isomorphism of topological $K$-vector spaces

$$\varprojlim_{m \in \mathbb{N}} H^i_{Y_I^-(\epsilon_m)}(\mathscr{F}^{\operatorname{rig}},\mathcal{E}_\lambda) \cong \varprojlim_{m \in \mathbb{N}} \hat{H}^i_{Y_I,m}.$$

The assertion is presented as an assumption and conjecture needed to control the Čech-computed rigid cohomology and compare it with algebraic and completed cohomology; the supplied source gives no resolution, so its status remains open.