In this paper, we would like to point out that in the scenario that the right-handed neutrinos suddenly gain some masses much larger than the temperature of the Universe at that time so that the washout effects for the lepton asymmetry generated from their decays can be neglected safely, the purely flavored leptogenesis scenario (in which the total CP asymmetries for the decays of the right-handed neutrinos are vanishing and the successful leptogenesis is realized by virtue of the flavor non-universality of the washout effects) cannot work in the usual way any more. For this problem, we put forward that the flavor non-universality of the conversion efficiencies from the flavored lepton asymmetries to the baryon asymmetry via the sphaleron processes may play a crucial role. And we will study if the requisite baryon asymmetry can be successfully reproduced from such a mechanism in the scenarios that the right-handed neutrino masses are hierarchical and nearly degenerate, respectively. A detailed study shows that this mechanism can be viable in both these two scenarios.