The conjecture on de Rham representations in analytic cohomology of the Lubin–Tate tower

Let $\rho_p:G_{\mathbb{Q}_p}\rightarrow\operatorname{GL}_2(E)$ be a continuous de Rham Galois representation, and let $B(\rho_p)$ denote the associated $p$-adic local Langlands representation. Let $\mathcal{M}_{LT,\infty}$ be the infinite-level Lubin–Tate tower, with analytic cohomology $H^1_{an}(\mathcal{M}_{LT,\infty},\mathcal{O}_{\mathcal{M}_{LT,\infty}})$ carrying an action of $\operatorname{GL}_2(\mathbb{Q}_p)$. Conjecture. There is a non-zero $\operatorname{GL}_2(\mathbb{Q}_p)$-equivariant injection

$$B(\rho_p)\hookrightarrow H^1_{an}(\mathcal{M}_{LT,\infty},\mathcal{O}_{\mathcal{M}_{LT,\infty}}).$$

This is proposed as a corrected version of the folklore conjecture because the relevant analytic cohomology representation is not admissible, preventing the authors from applying their localization method. The assertion is left as a conjecture for continuous de Rham representations.