High-pliability Fano hypersurfaces

We show that five of Reid's Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type $cA_n$ have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of $\mathbb{P}^2 \times \mathbb{P}^2$-type having Picard rank 2.