Rationally connected rational double covers of primitive Fano varieties

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb{P}}^{M+1}$ 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.