Baum–Connes vanishing conjecture for the fibrewise $L^2$-rho form

Let $\pi\colon M\to B$ be a smooth fibre bundle of compact manifolds, let $p\colon \widetilde M\to M$ be a normal covering with torsion-free deck-transformation group $\Gamma$, and let $\hat{\rho}_{(2)}(\mathcal{D}\kern-6.5pt/_{\hat g})$ denote the associated $L^2$-rho form for a metric $\hat g$ on the fibres. Write $\mathcal R^+(M/B)$ for the space of fibrewise positive-scalar-curvature metrics. Baum–Connes vanishing conjecture. If $\Gamma$ satisfies the Baum–Connes conjecture for the maximal $C^*$-algebra, then

$$[\hat{\rho}_{(2)}(\mathcal{D}\kern-6.5pt/_{\hat g})]=0$$

whenever $\hat g\in\mathcal R^+(M/B)$. This predicts vanishing of the cohomology class represented by the fibrewise $L^2$-rho form under the Baum–Connes hypothesis; the source gives no resolution status.