The cohomological comparison conjecture for algebraic double loop spaces of flag varieties

Let $\operatorname{Fl}_{n+1}$ be the flag variety, let $\beta$ be a strictly monotonic class, and write $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ for the corresponding algebraic double loop space and $\Omega^{2}_{\beta,\mathsf{top}}(\operatorname{Fl}_{n+1})$ for its topological counterpart. The inclusion map

$$\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})\hookrightarrow \Omega^{2}_{\beta,\mathsf{top}}(\operatorname{Fl}_{n+1})$$

can be compared with the rational cohomology of $U(n)$. Cohomological comparison conjecture. For strictly monotonic classes $\beta$, the inclusion map induces an isomorphism of rings

$$H^{*}(\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1}),{\mathbb{Q}})\cong H^{*}(U(n),{\mathbb{Q}}).$$

This conjecture asks whether the algebraic double loop spaces have the rational cohomology predicted by their topological counterparts. It is motivated by known agreement in the minimal strictly monotonic case and for infinitely large strictly monotonic classes, while finite non-minimal strictly monotonic classes can fail to have the homotopy type of $U(n)$ even rationally.