The omnioriented quasitoric representative conjecture for unitary equivariant bordism

Let $\mathcal{Z}_{2n}^{U}(T^n)$ be the group of $T^n$-equivariant unitary bordism classes of $2n$-dimensional unitary toric manifolds, let $\mathcal{Q}_*$ be the graded noncommutative ring generated by quasitoric pairs, and let $\mathcal{M}:\mathcal{Q}_*\longrightarrow\Xi_*$ be the homomorphism sending a quasitoric pair to its associated omnioriented quasitoric manifold. Omnioriented quasitoric representative conjecture. Each class of $\mathcal{Z}_{2n}^{U}(T^n)$ is represented by an omnioriented quasitoric manifold. Equivalently, $\mathcal{M}$ is surjective. Surjectivity is known for $n=1,2$, while the general case remains open.