Wilson's conjecture for omega-categorical locally nilpotent p-groups

A group is locally nilpotent if every finitely generated subgroup is nilpotent. A group is omega-categorical if its first-order theory has a unique countable model up to isomorphism, and a $p$-group is a group in which every element has order a power of the prime $p$.

Wilson's conjecture. Every locally nilpotent omega-categorical $p$-group is nilpotent.

Wilson's conjecture is a foundational problem concerning the interaction between model-theoretic symmetry and local-to-global nilpotency. The present paper proves a Lie-algebra analogue in the $3$-Engel characteristic-$5$ setting, but the group-theoretic conjecture stated here is not resolved in the supplied source.