Unstable relative Gromov–Lawson–Rosenberg conjecture

Let $(N,\partial N)$ be an $n$-dimensional compact spin manifold with boundary, and let $D$ be its Dirac operator. The relative higher index of $D$ is an element of $KO_n(\pi_1(N),\pi_1(\partial N))$.

Unstable relative Gromov–Lawson–Rosenberg conjecture. If the relative higher index of $D$ is zero in

$$KO_n(\pi_1(N),\pi_1(\partial N)),$$

then $N$ admits a metric of positive scalar curvature that is collared near $\partial N$.

This is a relative positive-scalar-curvature existence conjecture for compact spin manifolds with boundary. It is cited in the paper as an unstable relative form of the Gromov–Lawson–Rosenberg conjecture; the supplied text gives no resolution.