A counterexample to a conjecture of Plotnick on knot exteriors

We resolve a conjecture of S. Plotnick from 1983 concerning the homotopy type of four-dimensional knot complements. We show that the exteriors of two branched twist spins of the trefoil, which were shown by Plotnick not to be homotopy equivalent as boundary pairs and hence not homeomorphic, are in fact simple homotopy equivalent. The main ingredient of the proof is the construction of an explicit element of augmentation $49$ in the integral group ring of ${\mathrm{SL}}_2({\mathbb{F}}_5)$ which is a unit modulo the group norm and commutes with the element inducing the monodromy on the fundamental group.