The pessimistic conjecture on closed images of saturated submodules

Let $V$ be a $\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathcal{K}_\infty)$-representation, let $\mathcal{D}=\mathbf{D}^\dag_{\mathrm{rig}}(V)$, and let $\mathcal{D}^+$ be a saturated $\mathcal{R}$-submodule of $\mathcal{D}$. Consider the induced map

$$H^1(\mathcal{D}^+)\longrightarrow H^1(\mathcal{D})\cong H^1(\mathcal{K}_\infty,V).$$

Pessimistic conjecture. The image of this map is closed in the Banach-space topology of $H^1(\mathcal{K}_\infty,V)$. This weaker assertion is stated as sufficient for the paper's purposes, whereas the stronger canonical-topology conjecture is not known in full generality.