Comparison conjecture for étale Lubin–Tate $(\varphi_L,\Gamma_L)$-modules

Let $V$ be an $o_L$-linear representation of $G_L$, let $M_{\mathrm{LT}}=\mathbb{D}(V)$ be an étale $L$-analytic $(\varphi_L,\Gamma_L)$-module over $\mathbf{A}_L$, and let $M_{\mathrm{rig}}:=\mathbb{D}^{\dagger}_{\mathrm{rig}}(V)$. Write $\Lambda=o_L\llbracket\Gamma_L\rrbracket$, and let $\psi$ be the integral left inverse of $\varphi_L$. Comparison conjecture. The natural comparison map

$$D(\Gamma_L,L)\otimes_{\Lambda}M_{\mathrm{LT}}^{\psi=1}\xrightarrow{\operatorname{comp}}M_{\mathrm{rig}}^{\psi=1}$$

is surjective. This conjecture asks whether the $\psi=1$ vectors in the rigid analytic module are obtained from the integral étale module by extension of scalars to the distribution algebra; the supplied text gives no resolution, so it remains open.