Wilson conjecture for omega-categorical Lie algebras, the case 3-Engel characteristic 5

We prove a version of the Wilson conjecture for $\omega$-categorical $3$-Engel Lie algebras over a field of characteristic $5$: every $\omega$-categorical Lie algebra over ${\mathbb{F}}_5$ which satisfies the identity $[x,y^3]=0$ is nilpotent. We also include an extended introduction to Wilson’s conjecture: $\mathit{\text{every}}\omega \mathit{\text{-categorical}}\mathit{\text{locally}}\mathit{\text{nilpotent}}p\mathit{\text{-group}}\mathit{\text{is}}\mathit{\text{nilpotent}}$, and present variants of this conjecture and connections to local/global nilpotency problems (Burnside, Kurosh-Levitzki, Engel groups). No particular knowledge of model theory is assumed except basic notions of formulas and definable sets.