Recall that a subset $I$ of a ring $R$ is left (resp., right) $T$-nilpotent in case for every sequence $$a_1,a_2,\cdots $$ in $I$ there is an $n$ such that $a_1\cdots a_n=0$ (resp., $a_n\cdots a_1=0$). Every nilpotent ideal is left and right $T$-nilpotent.
Is there any characterization for rings $R$ in which the Jacobson radical is $T$-nilpotent?