  Different works have recently found an increase of the average X-ray-to-radio
luminosity ratio with redshift in the blazar population. We evaluate here
whether the inverse Compton interaction between the relativistic electrons
within the jet and the photons of the Cosmic Microwave Background (IC/CMB) can
explain this trend. Moreover, we test whether the IC/CMB model can also be at
the origin of the different space density evolutions found in X-ray and radio
selected blazar samples. By considering the best statistically complete samples
of blazars selected in the radio or in the X-ray band and covering a large
range of redshift (0.5$\lesssim$ z$\lesssim$ 5.5), we evaluate the expected
impact of the CMB on the observed X-ray emission on each sample and then we
compare these predictions with the observations. We find that this model can
satisfactorily explain both the observed trend of the X-ray-to-radio luminosity
ratios with redshift and the different cosmological evolutions derived from the
radio and X-ray band. Finally, we discuss how currently on-going X-ray
missions, like eROSITA, could help to further constrain the observed evolution
at even higher redshifts (up to z$\sim$6-7).
