Let F be a field and V a vector space over F. If G is a subgroup of G L(V, F), then we define the central dimension of G (denoted by centdim F G) as the F-dimension of the factor-space V /C V (G). In this paper, we continue the study of locally nilpotent linear groups satisfying the weak minimal or the weak maximal condition on their subgroups of infinite central dimension started in Kurdachenko et al. (Publ Mat 52:151-169, 2008).