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

We prove that a general Fano fibration $\pi \text{\hspace{0.17em}⁣}:V\to {\mathbb{P}}^1$, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on $V$ there are no other structures of a rationally connected fibration. The proof is based on the method of maximal singularities.