Perfectness conjecture for analytic $(\varphi_L,\Gamma_L)$-modules from $\mathcal{R}_L$

Let $L$ be a finite extension of $\mathbb{Q}_p$, let $K$ be a coefficient field containing $L$, and let $M_0$ be an $L$-analytic $(\varphi_L,\Gamma_L)$-module over $\mathcal{R}_L$. Let $C_{\Psi}(K\widehat{\otimes}_L M_0)$ denote the associated $\Psi$-cohomology complex, and let $D(\Gamma_L,K)$ be the distribution algebra of $\Gamma_L$. Perfectness conjecture. The complex

$$C_{\Psi}(K\widehat{\otimes}_L M_0)$$

is a perfect complex of $D(\Gamma_L,K)$-modules. This is the assertion needed to extend perfectness results from the cyclotomic setting to the Lubin–Tate analytic setting; the supplied text gives no resolution, so it remains open.