The Poincaré conjecture is true in the product of any graph with a disk

We prove that the only compact 3-manifold-with-boundary which has trivial rational homology, and which embeds in the product of a graph with a disk, is the 3-ball. This implies that no punctured lens space embeds in the product of a graph with a disk. It also implies our title. The proof relies on a general position argument which enables us to perform surgery.