We consider the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms.  Answering a question of Matsumoto, Adam Levine previously showed that the natural map from the smooth knot concordance group to this group is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order. This is joint work with Adam Levine and Tye Lidman.