Lubin-Tate and multivariable (, OK^) -modules in dimension 2

Let p be a prime number, K a finite unramified extension of Qₚ and F a finite extension of Fₚ. For any reducible two-dimensional representation of Gal (K/K) over F, we compute explicitly the associated \'etale (, OK^) -module DA^ () defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let be an admissible smooth representation of GL₂ (K) over F occurring in some Hecke eigenspaces of the mod p cohomology and be its underlying two-dimensional representation of Gal (K/K) over F. Assuming that is maximally non-split, we prove under some genericity assumption that the associated \'etale (, OK^) -module DA () defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to DA^ (). This extends the results of Breuil-Herzig-Hu-Morra-Schraen, where was assumed to be semisimple.