Gromov's compactness conjecture for smooth spin manifolds

Let $X$ be a smooth spin manifold. For a compact subset $V\subset X$ and $\rho>0$, let $N_\rho(V)$ denote the closed $\rho$-neighborhood of $V$ in $X$. Assume that there exists a non-complete Riemannian metric on $X$ with scalar curvature at least $1$ for every such $V$ and $\rho$, and let the $\sideset{}{^1}\varprojlim$ higher index of the Dirac operator of $X$ be the corresponding obstruction.

Gromov's compactness conjecture. If the $\sideset{}{^1}\varprojlim$ higher index of the Dirac operator of $X$ vanishes, then $X$ admits a complete Riemannian metric with scalar curvature at least $1$.

This is the precise formulation proposed in the paper after constructing negative examples caused by nonvanishing derived-limit index classes. Its resolution would give a positive answer to the compactness question under the stated vanishing hypothesis.