Let $α$ range over the set of compositions. Dual immaculate quasisymmetric functions $\mathfrak{S}_α^*$, introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki, provide a quasisymmetric analogue of Schur functions. They also constructed an indecomposable $0$-Hecke module $\mathcal{V}_α$ whose image under the quasisymmetric characteristic is $\mathfrak{S}_α^*$. In this paper, we prove that $\mathcal{V}_α$ admits a distinguished filtration with respect to the basis of Young quasisymmetric Schur functions. This result offers a novel representation-theoretic interpretation of the positive expansion of $\mathfrak{S}_α^*$ in the basis of Young quasisymmetric Schur functions. A key tool in our proof is Mason's analogue of the Robinson-Schensted-Knuth algorithm, for which we establish a version of Green's theorem. As an unexpected byproduct of our investigation, we construct an indecomposable $0$-Hecke module $\mathbf{Y}_α$ whose image under the quasisymmetric characteristic is the Young quasisymmetric Schur function $\hat{\mathscr{S}}_α$. Further properties of this module are also investigated. And, by applying a suitable automorphism twist to this module, we obtain an indecomposable $0$-Hecke module whose image under the quasisymmetric characteristic is the quasisymmetric Schur function $\mathscr{S}_α$.