Identify the infinite quasisymmetric Grassmannian with the Baker–Richter loop-space model

Establish that the infinite quasisymmetric Grassmannian $Q_{\infty}$ is isomorphic to the loop space $\Omega\Sigma\mathbb{C}P^{\infty}$ as a weak $H$-group.

Background

The paper constructs an ind-variety $Q_{\infty}$ as a direct limit of toric Richardson varieties in Grassmannians and equips it with a weak $H$-group structure. Its cohomology and homology are identified, respectively, with the Hopf algebras of quasisymmetric functions $QSym$ and noncommutative symmetric functions $NSym$.

Baker–Richter previously showed that the loop space $\Omega\Sigma\mathbb{C}P^{\infty}$ has cohomology $QSym$ and homology $NSym$ as Hopf algebras. The conjecture asks whether the geometric model constructed in the paper and the Baker–Richter topological model are actually isomorphic as weak $H$-groups, rather than merely sharing the same Hopf-algebraic invariants. The paper later repeats this conjecture in a more detailed form as Conjecture \ref{cj:homotopic2james}.