Lubin-Tate and multivariable $(\varphi,\mathcal{O}_K^{\times})$-modules in dimension 2

Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. For $\overline{\rho}$ any reducible two-dimensional representation of $\operatorname{Gal}(\overline{K}/K)$ over $\mathbb{F}$, we compute explicitly the associated \'etale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A^{\otimes}(\overline{\rho})$ defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let $\pi$ be an admissible smooth representation of $\operatorname{GL}_2(K)$ over $\mathbb{F}$ occurring in some Hecke eigenspaces of the mod $p$ cohomology and $\overline{\rho}$ be its underlying two-dimensional representation of $\operatorname{Gal}(\overline{K}/K)$ over $\mathbb{F}$. Assuming that $\overline{\rho}$ is maximally non-split, we prove under some genericity assumption that the associated \'etale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to $D_A^{\otimes}(\overline{\rho})$. This extends the results of Breuil-Herzig-Hu-Morra-Schraen, where $\overline{\rho}$ was assumed to be semisimple.