2 problems

Integral lifting conjecture. The resulting action of $U_f^\vee\otimes k$ on the special fiber lifts to an integral action of $U_f^\vee$.

Motivic action conjecture. There is a degree-shifting action of $U_f^\vee$ such that $H^\ast(X,\omega)_f$ is a rank-one module over $\bigwedge^\ast U_f^\vee$, generated by…