The NIP conjecture for omega-categorical rings

Let $R$ be a ring whose first-order theory is NIP, and suppose that $R$ is $\omega$-categorical, meaning that its theory has at most one countable model up to isomorphism. A ring is nilpotent-by-finite if it has a nilpotent ideal of finite index.

NIP conjecture for omega-categorical rings. Every $\omega$-categorical ring with NIP is nilpotent-by-finite.

The Jacobson radical of an omega-categorical ring is known to be nilpotent, but infinite omega-categorical semisimple rings show that omega-categorical rings need not be nilpotent-by-finite in general. The conjecture asks whether the additional NIP hypothesis forces nilpotence-by-finiteness and is presented as an open problem.