The electronic coupling to the longitudinal optical phonons has been recently used to interpret theoretically the electronic energy relaxation in quantum dots. In this theory the LO phonon modes served as a reservoir, on which the electron executes multiple scattering acts. Quite reasonably such a phonon subsystem is expected to be passive, namely, in a long-time limit of development, the whole system should be able to achieve such a stationary state, in which the statistical distributions of both subsystems, electron and phonons, do not change in time. We remind briefly that the recent approach to the relaxation in quantum dots has led to a non-passivity of such a reservoir. We remind as well the method of a partial elimination of the phonon overheating effect by using the Lang-Firsov transformation. Then we apply such a modified relaxation theory to the electronic relaxation at low electronic densities in quantum dots and come to conclusions concerning the role of e-LO scattering mechanism. We show that the modified theory of relaxation gives a plausible dependence of relaxation rate on temperature of sample. We also come to conclusions that at low levels of electronic excitation of quantum dot sample the electron-LO phonon mechanism can become relatively weak with respect to other relaxation mechanisms not supporting the electronic up-conversion effect.