Recently, the asymptotic behavior of three-dimensional anti-de Sitter (AdS) gravity with a topological mass term was investigated. Boundary conditions were given that were asymptotically invariant under the two dimensional conformal group and that included a falloff of the metric sufficiently slow to consistently allow pp-wave type of solutions. Now, pp waves can have two different chiralities. Above the chiral point and at the chiral point, however, only one chirality can be considered, namely, the chirality that has the milder behavior at infinity. The other chirality blows up faster than AdS and does not define an asymptotically AdS spacetime. By contrast, both chiralities are subdominant with respect to the asymptotic behavior of AdS spacetime below the chiral point. Nevertheless, the boundary conditions given in the earlier treatment only included one of the two chiralities (which could be either one) at a time. We investigate in this paper whether one can generalize these boundary conditions in order to consider simultaneously both chiralities below the chiral point. We show that this is not possible if one wants to keep the two-dimensional conformal group as asymptotic symmetry group. Hence, the boundary conditions given in the earlier treatment appear to be the best possible ones compatible with conformal symmetry. In the course of our investigations, we provide general formulas controlling the asymptotic charges for all values of the topological mass (not just below the chiral point).