5 problems

Agafonov–Ferapontov conjecture. Such a congruence always has an algebraic focal variety, possibly reducible and singular.

Generic $A_{\infty}$ conjecture. For an open and dense subset of $\mathcal{W}$, the umbilical point is of type $A_{\infty}$.

Generic $A_m$ conjecture. For a general $m\in\mathbb{N}$, an open and dense subset of $\mathcal{W}$ has non-degenerate umbilical points of type $A_m$.

Automatic-vanishing conjecture. For every $m\in\mathbb{N}$, the equation $W_{m0}=0$ holds automatically.

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