Lubin-Tate and multivariable $(\phi ,{\mathcal{O}}_K^{\times })$-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 $\bar{\rho }$ any reducible two-dimensional representation of $\mathrm{Gal}(\bar{K}\mathrm{/}K)$ over $\mathbb{F}$, we compute explicitly the associated étale $(\phi ,{\mathcal{O}}_K^{\times })$-module $D_A^{\otimes }(\bar{\rho })$ defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let $\pi$ be an admissible smooth representation of ${\mathrm{GL}}_2(K)$ over $\mathbb{F}$ occurring in some Hecke eigenspaces of the mod $p$ cohomology and $\bar{\rho }$ be its underlying two-dimensional representation of $\mathrm{Gal}(\bar{K}\mathrm{/}K)$ over $\mathbb{F}$. Assuming that $\bar{\rho }$ is maximally non-split, we prove under some genericity assumption that the associated étale $(\phi ,{\mathcal{O}}_K^{\times })$-module $D_A(\pi )$ defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to $D_A^{\otimes }(\bar{\rho })$. This extends the results of Breuil-Herzig-Hu-Morra-Schraen, where $\bar{\rho }$ was assumed to be semisimple.