Radicals around Köthe's problem

Radicals $\gamma$ will be studied for which the condition "$A[x] \in\gamma$ for all nil rings $A$" is equivalent to the positive solution of Köthe's Problem ($A[x]$ is Jacobson radical for all nil rings $A$, in Krempa's formulation). The closer $\gamma$ is to the Jacobson radical, the better approximation of the positive solution is obtained. Seeking, however, for a negative solution, possibly large radicals $\gamma$ are of interest. In this note such large radicals will be studied.