Cross-population radio galaxies (FRII-LERG, i.e., powerful radio sources with an inefficient engine) are a powerful tool to investigate the long-standing relation between accretion and ejection. Indeed, they break the classical one-to-one correspondence between accretion mode and radio morphology by separating the effects of optical and radio classifications. To shed light on the nature of FRII-LERG, we present a comparative study of the X-ray and CO properties of a sample of nearly a hundred sources classified in the radio (FRI/FRII) and optical (HERG/LERG) bands. X-rays probe the AGN's innermost regions (sub-pc scale) and allow us to estimate the accretion rate free from contamination effects accurately. CO emission lines represent the best tracer of the molecular ISM, considered a fuel for AGN activity. This systematic and uniform study will help us verify if FRII-LERG are evolved sources, i.e., they now miss the cold fueling material that made them shine in the past