The finite-by-null conjecture for omega-categorical pseudofinite rings

Let $R$ be a ring that is both omega-categorical, meaning that its theory has a unique countable model up to isomorphism, and pseudofinite, meaning that it is infinite and every first-order sentence true in $R$ has a finite model. A ring is null if $xy=0$ for all $x,y\in R$. It is finite-by-null if it has a finite two-sided ideal $J$ such that the quotient by $J$ is null. The finite-by-null conjecture. The ring $R$ has a definable two-sided ideal $I$ of finite index which is finite-by-null; that is, $I$ has a finite two-sided ideal $J$ such that $I/J$ is null. This conjecture is presented as a ring-theoretic analogue of the preceding group conjecture; the supplied text does not state a resolution.