Uniform vector bundles over ${\mathbb{P}}^4$

There is a long-standing conjecture which states that every uniform algebraic vector bundle of rank $r<2n$ on the $n$-dimensional projective space ${\mathbb{P}}^n$ over an algebraically closed field of characteristic $0$ is homogeneous. This conjecture is valid for $n\le 3$. In this paper, we classify all uniform vector bundles of rank $r<8$ over ${\mathbb{P}}^4$ and show that the conjecture holds for $n=4$.