  Let $K$ and $L$ be two convex bodies in ${\mathbb R^5}$ with countably many
diameters, such that their projections onto all $4$ dimensional subspaces
containing one fixed diameter are directly congruent. We show that if these
projections have no rotational symmetries, and the projections of $K,L$ on
certain 3 dimensional subspaces have no symmetries, then $K=\pm L$ up to a
translation. We also prove the corresponding result for sections of star
bodies.
