The solid Fano hypersurface conjecture for index at least two

Let $X$ be a quasi-smooth Fano hypersurface, and let $\iota_X$ denote its Fano index. A Fano variety is solid if it admits no birational map to a Mori fibre space over a positive-dimensional base. Solid Fano hypersurface conjecture. The hypersurface $X$ is solid if and only if it belongs to one of the families $\textnumero\,100$, $\textnumero\,101$, $\textnumero\,102$, $\textnumero\,103$, $\textnumero\,110$. The paper proves birational non-rigidity for the other quasi-smooth Fano threefold hypersurfaces with index at least two, providing evidence for this conjectural classification of solid Fano varieties; the converse and the solidity of the listed families remain to be established.