On the rank of the multivariable (φ,OK×)-modules associated to mod p representations of GL₂(K)

Let p be a prime number, K a finite unramified extension of Qₚ and F a finite extension of Fₚ. For π an admissible smooth representation of GL₂(K) over F satisfying certain multiplicity-one properties, we compute the rank of the associated \'etale (φ,OK×)-module DA(π) defined by Breuil-Herzig-Hu-Morra-Schraen, extending their results.