The Baker-Richter spectrum as cobordism of quasitoric manifolds

Baker and Richter construct a remarkable $A_{\mathrm{\infty }}$ ring-spectrum $M\mathrm{\Xi }$ whose elements possess characteristic numbers associated to quasisymmetric functions; its relations, on one hand to the theory of noncommutative formal groups, and on the other to the theory of omnioriented (quasi)toric manifolds [in the sense of Buchstaber, Panov, and Ray], seem worth investigating.