The generalized Lichnerowicz conjecture

Let $(M,g)$ be a pseudo-Riemannian manifold. Its conformal group is denoted by $\mathrm{Conf}^\uparrow(M,g)$, and this group is essential if there is no metric conformal to $g$ for which it acts by isometries. Generalized Lichnerowicz conjecture. If $(M,g)$ has essential conformal group, then $(M,g)$ is conformally flat. The conjecture is false in several settings, including compact signatures $(p,q)$ with $p,q\geq 2$ and non-compact Lorentzian geometry, but remains open for compact Lorentzian manifolds, where it is known under additional assumptions.