Overconvergence conjecture for F-analytic representations

Let $K$ be the finite extension of the maximal unramified extension of a finite extension of $\mathbb{Q}_p$ considered in the paper, let $F$ be its coefficient field, and let $V$ be an $F$-linear representation of $G_K$. The representation $V$ is $F$-analytic when its associated $(\varphi_q,\Gamma_K)$-module satisfies the paper's $F$-analyticity condition, and it is overconvergent when it arises from an overconvergent $(\varphi_q,\Gamma_K)$-module. Overconvergence conjecture. If $V$ is $F$-analytic, then it is overconvergent. This extends the known case of crystalline representations, for which the assertion follows from work of Kisin and Ren; whether it holds for all $F$-analytic representations remains open.