The boundary of the refined Kingman graph

We introduce the refined Kingman graph $\mathbb D$ whose vertices are indexed by the set of compositions of positive integers and multiplicity function reflects the Pieri rule for quasisymmetric monomial functions. We show that the Martin boundary of $\mathbb D$ can be identified with the space $\Omega$ of all sets of disjoint open subintervals of $[0,1]$ and coincides with the minimal boundary of $\mathbb D$.