  We introduce a geometric operation, which we call the relative Whitney trick,
that removes a single double point between properly immersed surfaces in a
$4$-manifold with boundary. Using the relative Whitney trick we prove that
every link in a homology sphere is homotopic to a link that is topologically
slice in a contractible topological $4$-manifold. We further prove that links
in homology spheres are equivalent to links in $S^3$ up to any level of the
Whitney tower filtration that approximates link concordance. Finally, we
explore the minimum Gordian distance from a link in $S^3$ to a homotopically
trivial link. Extending this notion to links in homology spheres, we use the
relative Whitney trick to make explicit computations for 3-component links and
establish bounds in general.
