Splitting links by integer homology spheres

For every $n\ge 3$, we construct 2-component links in $S^{n+1}$ that are a split by an integer homology $n$-sphere, but not by $S^n$. In the special case $n=3$, i.e. that of 2-links in $S^4$, we produce an infinite family of links $L_{\mathrm{\ell }}$ and of integer homology spheres $Y_{\mathrm{\ell }}$ such that the link $L_{\mathrm{\ell }}$ is (topologically or smoothly) split by $Y_{\mathrm{\ell }}$ and by no other integer homology sphere in the family.