  We show that for a Zariski general hypersurface $V$ of degree $M+1$ in
${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers
$X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group,
where $X$ is a rationally connected variety. In particular, there are no
rational maps $X\dashrightarrow V$ 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.
