Non-existence of low rank Ulrich bundles on Veronese varieties

We show that Veronese varieties of dimension n ≥ 4 do not carry any Ulrich bundles of rank r ≤ 3. In order to prove this, we prove that a Veronese embedding of a complete intersection of dimension m ≥ 4, which if $m=4$ is either P⁴ or has degree d ≥ 2 and is very general and not of type $(2), (2,2)$, does not carry any Ulrich bundles of rank r ≤ 3.