Fibrewise homotopy invariance conjecture for the signature $L^2$-rho form

Let $\pi\colon M\to B$ and $\theta\colon N\to B$ be smooth fibre bundles of compact manifolds over the same base, with normal $\Gamma$-coverings $p\colon \widetilde M\to M$ and $q\colon \widetilde N\to N$, where $\Gamma$ is a discrete group. A $\Gamma$-fibrewise homotopy equivalence is a fibrewise homotopy equivalence compatible up to homotopy with the classifying maps to $B\Gamma$. Let $\mathcal{\widetilde D}^{\mathrm{sign}}_{M/B}$ and $\mathcal{\widetilde D}^{\mathrm{sign}}_{N/B}$ be the lifted families of signature operators, and let $\hat\rho_{(2)}$ denote their fibrewise $L^2$-rho forms. Fibrewise homotopy invariance conjecture. Assume that $\Gamma$ is torsion-free and satisfies the Baum–Connes conjecture for the maximal $C^*$-algebra. If $h$ is an orientation-preserving $\Gamma$-fibrewise homotopy equivalence between $(\pi,p)$ and $(\theta,q)$, and both lifted signature families have smooth spectral projections and Novikov–Shubin invariants greater than $3(\mathop{\rm \dim} B+1)$, then

$$[\hat\rho_{(2)}(\mathcal{\widetilde D}^{\mathrm{sign}}_{M/B})]=[\hat\rho_{(2)}(\mathcal{\widetilde D}^{\mathrm{sign}}_{N/B})]\in H^*_{\mathrm{dR}}(B).$$

This predicts that the fibrewise $L^2$-rho cohomology class of the signature family is invariant under the stated geometric equivalence and analytic regularity assumptions; the source gives no resolution status.