The relative index comparison conjecture of Chang, Weinberger and Yu

Let $\overline{\Omega}$ be a compact manifold with boundary and fundamental group $\Gamma$, let $\partial\Omega$ have fundamental group $\Gamma'$, and let $\overline{\Omega}^{\sim}$ be the universal cover of $\overline{\Omega}$. Let $D^*(\overline{\Omega}^{\sim},\partial(\Omega^{\sim}))^{(\Gamma,\Gamma')}$ be an equivariant refinement of the relative structure algebra.

Relative index comparison conjecture. There exists a map

$$K_*\left(D^*(\overline{\Omega}^{\sim},\partial(\Omega^{\sim}))^{(\Gamma,\Gamma')}\right)\longrightarrow K_*\left(C^*_{\max}(\Gamma,\Gamma')\right)$$

that maps the relative index class to the relative index of Chang, Weinberger and Yu.

The source calls this a vague conjecture concerning the relationship between two relative index theories and gives no resolution status.