General equivariant higher signature homotopy invariance conjecture

Let $M$ be a closed oriented manifold with an effective $S^1$-action. For each connected component $F$ of $M^{S^1}$, assume the hypotheses on $\Gamma_F$, its smooth subalgebra, and the cyclic cocycle extension stated in the paper, and define $\langle\sigma_{S^1}(M),[\tau]\rangle$ by the orbifold integrals and higher eta-invariants. General equivariant higher signature conjecture.

$$\langle \sigma_{S^1}(M), [\tau] \rangle$$

is an $S^1$-homotopy invariant of $M$. The construction is independent of the choices of invariant metric and cutoff function under the stated assumptions; the conjecture asserts the remaining $S^1$-homotopy invariance.