Genomic Schur functions were introduced by Pechenik and Yong in connection with the $K$-theory of Grassmannians. Pechenik proved that genomic Schur functions admit a positive expansion in the basis of fundamental quasisymmetric functions and, for partitions with two parts, a positive expansion in the Schur basis. Later, Kim and Yoo constructed $0$-Hecke modules associated with genomic Schur functions and conjectured that the latter expansion admits a representation-theoretic interpretation in terms of $0$-Hecke modules. In this paper, we prove the conjecture of Kim and Yoo, thereby obtaining a representation-theoretic interpretation of the Schur expansion in the two-row case.