The topological concordance conjecture for links in homology spheres

Let $K\subseteq Y$ be a link in a homology sphere. A homology cobordism $W$ from $Y$ to $S^3$ is a cobordism inducing homology isomorphisms from its boundary components. Two links cobound a locally flat embedded annulus concordance when they cobound a disjoint union of locally flat embedded annuli in the cobordism. Topological concordance conjecture. There is a link $J$ in $S^3$ and a homology cobordism $W$ from $Y$ to $S^3$ in which $K$ and $J$ cobound a disjoint union of locally flat embedded annuli. Moreover, $W$ can be chosen to be simply connected. The conjecture proposes a contrast with the smooth setting, where it is false: Levine proved the existence of knots in homology spheres with no such smooth concordance to a knot in $S^3$, and subsequent work strengthened that result. The present paper gives evidence for the topological version and proves related statements for the Whitney tower and solvable filtrations.