  Two knots are homology concordant if they are smoothly concordant in a
homology cobordism. The group $\hat{\mathcal{C}}_{\mathbb{Z}}$ (resp.
$\mathcal{C}_{\mathbb{Z}}$) was previously defined as the set of knots in
homology spheres that bounds homology balls (resp. in $S^3$), modulo homology
concordance. We prove $\hat{\mathcal{C}}_{\mathbb{Z}} /
\mathcal{C}_{\mathbb{Z}}$ contains a $\mathbb{Z}^{\infty}$ subgroup. We
construct our family of examples by applying the filtered mapping cone formula
to $L$--space knots, and prove linear independence with the help of the
connected knot complex.
