Izumiya's Lagrangian stability conjecture for generic Blaschke affine normal congruences

A 3-parameter line congruence in $\mathbb{R}^4$ is a smooth map

$$F_{(\bm{x},\bm{\xi})}:U\times I\longrightarrow\mathbb{R}^4,\qquad F_{(\bm{x},\bm{\xi})}(u,t)=\bm{x}(u)+t\bm{\xi}(u),$$

where $U\subset\mathbb{R}^3$ is open, $I$ is an open interval, $\bm{x}$ parametrizes a reference hypersurface, and $\bm{\xi}$ parametrizes a director hypersurface. A Blaschke affine normal congruence is the congruence associated with the Blaschke vector field of a non-degenerate hypersurface in $\mathbb{R}^4$.

Izumiya's Lagrangian stability conjecture. Germs of generic Blaschke affine normal congruences at any point are Lagrangian stable.

The conjecture concerns the generic singularities of Blaschke affine normal congruences and their interpretation through Lagrangian singularity theory. The paper proves the classification of the generic singularities of Blaschke exact normal congruences and Blaschke normal congruences, thereby providing a positive answer to this conjecture.