Open null ideal conjecture for small profinite rings

Let $R$ be a small profinite ring, meaning a profinite ring whose automorphism group respecting a distinguished countable inverse system has only countably many orbits on $n$-tuples for every $n<\omega$. An ideal is null if its product with itself is zero.

Null ideal conjecture. A small profinite ring has an open null ideal.

This conjecture is presented as a further structural question about small profinite rings and is motivated by results on small profinite rings and groups. The supplied text gives no resolution status, so it remains open in this record.