Let $R$ be a ring with identity, and $e^2=e\in R$ such that both $eJe$ and $(1-e)J(1-e)$ are nil, where $J=J(R)$ is the Jacobson radical of $R$. When $R$ is commutative, it is easy to see that $J$ is nil. Indeed, the sum $ej+(1-e)j=j$ is nilpotent, for every $j\in J$ in the commutative setting. I guess that if Koethe's Conjecture holds for $R$, then $J$ is nil too. (The conjecture is true for the commutative rings.)
Any suggestion/help is appreciated in advance!