Density of pullback morphisms in the stable real Grassmannian ring

Prove or disprove that the pullback morphisms \(H(G(k,m_2))\to H(G(k,m_1))\) associated with the inclusions \(G(k,m_1)\hookrightarrow G(k,m_2)\) have dense image, and consequently characterize the image in the inverse-limit algebra \(H(G(k,\infty))\).

Background

For fixed $k$, the paper forms an inverse system of probabilistic intersection rings $H(G(k,m))$ using pullbacks induced by inclusions of real Grassmannians. Unlike the projective and complex-projective cases, these pullback maps do not appear to be surjective.

The authors formulate an explicit conjecture that the pullback maps nevertheless have dense image. Establishing this would provide important information about the stable probabilistic Schubert calculus and the structure of the inverse-limit algebra.