The relative Novikov conjecture for manifolds with boundary

Let $(M,\partial M)$ be a compact oriented smooth manifold with boundary, and let $G$ and $\Gamma$ be groups associated to the boundary and manifold through a homomorphism $h:G\to\Gamma$. Relative higher signatures are obtained by pairing the relative $L$-classes with classes $(\xi,\eta)\in H^*(BG,B\Gamma)$. Relative Novikov conjecture. All relative higher signatures are invariant under orientation-preserving homotopy equivalences of pairs. More precisely, if $\phi:(M,\partial M)\to(N,\partial N)$ is such an equivalence, then

$$\int_M \mathcal L_M\cup\psi_M^*(\xi)-\int_{\partial M}\mathcal L_{\partial M}\cup\psi_{\partial M}^*(\eta)=\int_N\mathcal L_N\cup\psi_N^*(\xi)-\int_{\partial N}\mathcal L_{\partial N}\cup\psi_{\partial N}^*(\eta)$$

for all $(\xi,\eta)\in H^*(BG,B\Gamma)$, with compatible classifying maps. This is the relative analogue of the Novikov conjecture for closed manifolds. The paper proves it under geometric conditions via maximal or reduced strong relative Novikov conjectures; the general assertion is presented as a conjecture.