Birationally rigid hypersurfaces with quadratic singularities of low rank

It is shown that hypersurfaces of degree $M$ in ${\mathbb P}^M$, $M\geqslant 5$, with at most quadratic singularities of rank at least 3, satisfying certain conditions of general position, are birationally superrigid Fano varieties and the complement to the set of such hypersurfaces is for $M\geqslant 8$ of codimension at least ${M-1 \choose 2} + 1$ with respect to the natural parameter space.