On bodies with congruent sections or projections

In this paper, we construct two convex bodies $K$ and $L$ in $\mathbb{R}^n$, $n\geq 3$, such that their projections $K|H$, $L|H$ onto every subspace $H$ are congruent, but nevertheless, $K$ and $L$ do not coincide up to a translation or a reflection in the origin. This gives a negative answer to an old conjecture posed by Nakajima and S\"uss.