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

Let $L$ be a finite extension of $\mathbb{Q}_p$, let $K$ be a coefficient field containing $L$, and let $V$ be an $L$-analytic $o_L$-linear representation of $G_L$. Let $M$ be the completed base change to $K$ of the $(\varphi_L,\Gamma_L)$-module over $\mathcal{R}_L$ attached to $V$, let $\mathbb{D}^{\dagger}(V)$ denote its overconvergent module, and put $\Lambda=o_L\llbracket\Gamma_L\rrbracket$. The operator $\psi_{\mathrm{LT}}$ is the Lubin–Tate left inverse of $\varphi_L$. Comparison conjecture. The natural map

$$D(\Gamma_L,K)\otimes_{\Lambda}\mathbb{D}^{\dagger}(V)^{\psi_{\mathrm{LT}}=1}\to M^{\psi_{\mathrm{LT}}=1}$$

is surjective. This predicts that the $\psi_{\mathrm{LT}}=1$ vectors in the analytic $(\varphi_L,\Gamma_L)$-module are generated after distribution-algebra base change from the overconvergent ones; the source provides no resolution, so the conjecture remains open.