  Let $\widehat{\mathcal{C}}_\mathbb{Z}$ be the group consists of manifold-knot
pairs $(Y,K)$ modulo homology concordance, where $Y$ is an integer homology
sphere bounding an integer homology ball, and let $\mathcal{C}_\mathbb{Z}$ be
the subgroup consisting of pairs $(S^3,K)$. Dai-Hom-Stoffregen-Truong show that
the quotient group
${\widehat{\mathcal{C}}_\mathbb{Z}}/{\mathcal{C}_\mathbb{Z}}$ admits a
$\mathbb{Z}^\infty$-summand. In this paper, we improve the result by showing
that there exists a family $\{(Y,K_m)\}_{m>1 }$ generating the
$\mathbb{Z}^\infty$-summand where $Y$ is the boundary of a smooth contractible
$4$-manifold. In fact, we give a $\mathbb{Z}$-count of such families.
  The examples are constructed using a family of knots obtained by blowing down
a component of a two-bridge link. They are studied in Jonathan Hales's thesis.
Using the algorithm due to Ozsv\'{a}th, Szab\'{o} and Hales we give a
classification of the knot Floer homology of a larger family of such knots,
that might be of independent interest.
