Birationally rigid varieties with a pencil of double Fano covers. I

The general Fano fibration $\pi\colon V\to\mathbb P^1$ the fibre of which is a double Fano hypersurface of index 1 is proved to be birationally superrigid, provided it is sufficiently twisted over the base. In particular, there exist on $V$ no other structures of a rationally convex fibration. The proof is based on the method of maximal singularities.