Rational analytic Novikov conjecture

Let $\Gamma$ be a discrete group, let $E\Gamma$ be the universal space for a free $\Gamma$-action, and let $e$ be the evaluation map from the equivariant localization algebra to the equivariant Roe algebra. Define

$$K_*(C^*_L(E\Gamma)^\Gamma)=\varinjlim_X K_*(C^*_L(X)^\Gamma),\qquad K_*(C^*(E\Gamma)^\Gamma)=\varinjlim_X K_*(C^*(X)^\Gamma),$$

where the limits range over locally compact, $\Gamma$-equivariant, $\Gamma$-cocompact subsets $X$ of $E\Gamma$.

Analytic Novikov conjecture. The map $e$ induces an injection

$$e_*:K_*(C^*_L(E\Gamma)^\Gamma)\longrightarrow K_*(C^*(E\Gamma)^\Gamma).$$

In the rational version, this map remains injective after tensoring with $\mathbb Q$.

The strong Novikov conjecture implies this analytic form of the Novikov conjecture. The paper presents the rational injectivity statement as conjectural.