  In a groundbreaking work A. Levine proved the surprising result that there
exist knots in homology spheres which are not smoothly concordant to any knot
in $S^3$, even if one allows for concordances in homology cobordisms. Since
then subsequent works due to Hom-Levine-Lidman and Zhou have strengthened this
result showing that there are many knots in homology spheres which are not
smoothly concordant to knots in $S^3$. In this paper we present evidence that
the opposite is true topologically. We study the Whitney tower filtration of
concordance due to Cochran-Orr-Teichner and prove that modulo any term in this
filtration every knot (or link) in a homology sphere is equivalent to a knot
(or link) in $S^3$. As an application we recover the main result of
[Davis2019], namely that the solvable filtration similarly fails to distinguish
links in homology spheres from links in $S^3$.
