Lubin-Tate and multivariable (φ,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 ρ̄ any reducible two-dimensional representation of Gal(K̄/K) over F, we compute explicitly the associated \'etale (φ,OK×)-module DA⊗(ρ̄) defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let π be an admissible smooth representation of GL₂(K) over F occurring in some Hecke eigenspaces of the mod p cohomology and ρ̄ be its underlying two-dimensional representation of Gal(K̄/K) over F. Assuming that ρ̄ is maximally non-split, we prove under some genericity assumption that the associated \'etale (φ,OK×)-module DA(π) defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to DA⊗(ρ̄). This extends the results of Breuil-Herzig-Hu-Morra-Schraen, where ρ̄ was assumed to be semisimple.