Signature-index invariance under oriented homotopy equivalence at infinity

Let $M_1$ and $M_2$ be complete, non-compact, connected, oriented Riemannian manifolds, and let $f\colon M_1\to M_2$ be a sufficiently well-behaved map that is an oriented homotopy equivalence near infinity. Let $E\to M_2$ be a flat bundle of Hilbert $A$-modules for a $C^*$-algebra $A$. Denote by $D^{\operatorname{sgn}}_E$ and $D^{\operatorname{sgn}}_{f^*E}$ the corresponding twisted signature operators, and by $\operatorname{ind}$ their large-scale indices.

Signature-index invariance conjecture. The large-scale indices should coincide:

$$f_*\bigl(\operatorname{ind}(D^{\operatorname{sgn}}_{f^*E})\bigr)=\operatorname{ind}(D^{\operatorname{sgn}}_E)\in K_*(C^*(M_1;A)).$$

The source explicitly says that the notions of “sufficiently well behaved” and “homotopy equivalence at infinity” still need to be made precise. The conjecture is intended as a large-scale analogue of homotopy invariance for the signature operator.