The differential equivariant K-theory pushforward conjecture

Let $X$ be a compact odd-dimensional Riemannian spin $\Gamma$-manifold, let $V$ be a $\Gamma$-equivariant vector bundle with invariant compatibly unitary connection $\nabla^V$, and let $D_V$ be the Dirac operator on the twisted spinor bundle $S_X\otimes V$. Let $\check{x}\in \check{K}^0_\Gamma(X)$ be the differential equivariant $K$-theory class represented by $(V,\nabla^V,0)$. The pushforward satisfies

$$\wp_\Gamma\left(\check{x}\right)=\xi_\Gamma(D_V)$$

Differential equivariant pushforward conjecture. The equality holds as class functions modulo $R(\Gamma)$. This conjectures that the index-theoretic description of the ordinary equivariant $K$-theory pushforward extends to differential equivariant $K$-theory, identifying the pushforward with the reduced equivariant eta invariant. It has been proved in certain special cases.