  The $0$-surgeries of two knots $K_1$ and $K_2$ are homology cobordant rel
meridians if there exists a $\mathbb{Z}$-homology cobordism $X$ between them
such that the two knot meridians are in the same homology class in
$H_{1}(X,\mathbb{Z})$. In this paper, we give a pair of rationally slice knots
which are not smoothly concordant but whose $0$-surgeries are homology
cobordant rel meridians. One knot in the pair is the figure eight knot, which
has concordance order two; all previous examples of such pairs of knots are
infinite order.
