Vanishing conjecture for higher localised $\widehat{A}$-genera

Let $\Gamma$ be a discrete group acting properly and cocompactly on a connected spin manifold $M$. Let $\underline{B}\Gamma$ denote the classifying space for proper actions, let $f\colon M/\Gamma\to\underline{B}\Gamma$ be the classifying map, and let $\omega\in\Omega^m(M)$ be a $\Gamma$-invariant lift of a closed differential form on the orbifold $M/\Gamma$ whose cohomology class is $f^*[\alpha]$ for some $\alpha\in Z^m(\underline{B}\Gamma,\mathbb{R})$. For each $g\in\Gamma$, let $M^g$ be the fixed-point submanifold, let $Z^g$ be the centraliser of $g$, and let $c^g$ be a cut-off function for the proper cocompact action of $Z^g$ on $M^g$. Define

$$\widehat A_{g}(M,\omega)\coloneqq\int_{M^g}c^g\cdot\frac{\widehat A(M^g)\cdot\omega|_{M^g}}{\det(1-g e^{-R^{\mathcal{N}}/2\pi i})^{1/2}}.$$

Vanishing conjecture for higher localised $\widehat{A}$-genera. If $M$ admits a $\Gamma$-invariant Riemannian metric of positive scalar curvature, then for every closed $m$-form $\omega$ constructed as above and every $g\in\Gamma$,

$$\widehat A_{g}(M,\omega)=0.$$

This is the proposed positive-scalar-curvature obstruction for proper cocompact actions, extending the torsion-free setting and incorporating fixed-point contributions from elements of $\Gamma$; the source gives no evidence of a resolution.