Nilpotence conjecture for the Jacobson radical of 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 $n<\omega$. Let $J(R)$ denote its Jacobson radical.

Nilpotence conjecture. The Jacobson radical $J(R)$ is nilpotent. In particular, $R$ has an open nilpotent ideal.

The conjecture concerns the structure of the radical in small profinite rings. It is motivated by the known result that $J(R)$ is open for such rings; the paper proves nilpotence for the Jacobson radical of weakly locally finite profinite rings, but the stated conjecture for all small profinite rings remains open.