Relative Gromov–Lawson–Rosenberg conjecture

Let $(N,\partial N)$ be a compact spin manifold with boundary, let $n=\dim N$, and let $D$ be its Dirac operator. Write $C^*(\pi_1(N),\pi_1(\partial N))$ for the relative group $C^*$-algebra and let the relative higher index of $D$ lie in

$$KO_n(C^*(\pi_1(N),\pi_1(\partial N))).$$

Relative Gromov–Lawson–Rosenberg conjecture. If the relative higher index of $D$ is zero, then $N$ admits a metric of positive scalar curvature that is collared near $\partial N$. This is the proposed relative analogue of the closed-manifold Gromov–Lawson–Rosenberg conjecture; the supplied text gives no resolution status.