The vanishing conjecture for Wodzicki-Chern classes of pseudodifferential operator bundles

Let $\Psi{\rm DO}_0^*$ be the group of invertible zeroth-order classical pseudodifferential operators, and let $\mathcal E$ be a $\Psi{\rm DO}_0^*$-bundle over a base manifold admitting a partition of unity. The Wodzicki-Chern classes $c_k^{\rm w}(\mathcal E)$ are defined using the Wodzicki residue applied to powers of the curvature. Wodzicki-Chern vanishing conjecture. The Wodzicki-Chern classes vanish on any $\Psi{\rm DO}_0^*$-bundle over a base manifold admitting a partition of unity. The preceding theorem proves vanishing when the structure group reduces to the subgroup of operators with identity leading symbol; the conjecture asserts the corresponding statement for arbitrary invertible zeroth-order pseudodifferential-operator bundles.