Is problem of finiteness in the sense of von Neumann field FG algebras solved for any class of groups?

Question

there is an open problem: Does for every field $F$ of characteristic prime and every group $G$, the group ring $FG$ finite in the sens of von Neumann? Is this problem solved for any class of groups?

I ask because for example Kaplansky's Conjecture is solved for torsionfree noetherian groups and fields of characteristic 0.