Gamma-ray luminosities of some quasar-associated blazars imply jet powers reaching values comparable to the accretion power even if assuming very strong Doppler boosting and very high efficiency of gamma-ray production. With much lower radiative efficiencies of protons than of electrons, and the recent reports of very strong coupling of electrons with shock-heated protons indicated by Particle-in-Cell (PIC) simulations, the leptonic models seem to be strongly favored over the hadronic ones. However, the electron-proton coupling combined with the ERC (External-Radiation-Compton) models of gamma-ray production in leptonic models predict extremely hard X-ray spectra, with energy indices about 0. This is inconsistent with the observed 2-10 keV slopes of blazars, which cluster around an index value of 0.6. This problem can be resolved by assuming that electrons can be cooled down radiatively to non-relativistic energies, or that blazar spectra are entirely dominated by the SSC (Synchrotron-Self Compton) component up to at least 10 keV. Here, we show that the required cooling can be sufficiently efficient only at distances r < 0.03pc. SSC spectra, on the other hand, can be produced roughly co-spatially with the observed synchrotron and ERC components, which are most likely located roughly at a parsec scale. We show that the dominant SSC component can also be produced much further than the dominant synchrotron and ERC components, at distances larger than 10 parsecs. Hence, depending on the spatial distribution of the energy dissipation along the jet, one may expect to see gamma-ray/optical events with either correlated or uncorrelated X-rays. In all cases the number of electron-positron pairs per proton is predicted to be very low. The direct verification of the proposed SSC scenario requires sensitive observations in the hard X-ray band which is now possible with the NuSTAR satellite. Show Partial Abstract