In the paper, Homology cylinders: an enlargement of the mapping class group, of J.Levine, I have a question in the proof of Theorem 4. The theorem says that there's injection $\hat\Phi$ from the concordance group $\mathcal{S}_g^\mathrm{fr}$ of framed $g $-string links to the homology cobordism group $\mathcal{H}_g$ of homology cylinders over $\Sigma_{g,1}$.

To prove this, he showed there is a map $\rho$ from a subset of $\mathcal{H}_g$ (containing the image of $\hat\Phi$) to $\mathcal{S}_g^{\mathrm{fr}}$, which gives $\rho \circ \hat\Phi = \mathrm{id}$.

But I cannot check the well-definedness of $\rho$, i.e., that if two are homology cobordant, then the images via $\rho$ are concordant.

Could anyone help me? Thank you.