We show that for a Zariski general hypersurface V of degree M+1 in PM+1 for M⩾5 there are no Galois rational covers X⇢V of degree d⩾2 with an abelian Galois group, where X is a rationally connected variety. In particular, there are no rational maps X⇢V of degree 2 with X rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties.