QCD ghost dark energy cannot (even roughly) explain the main features of the accepted cosmological paradigm

We explore the whole phase space of the so called Veneziano/QCD ghost dark energy models where the dynamics of the inner trapping horizon is ignored and also the more realistic models where the time-dependence of the horizon is taken into consideration. We pay special attention to the choice of phase space variables leading to bounded and compact phase space so that no critical point of physical interest is missing. It is demonstrated that ghost dark energy is not a suitable candidate to explain the presently accepted cosmological paradigm since no critical point associated with matter dominance is found in the physical phase space of the model. A transient stage of matter dominance -- responsible for the observed amount of cosmic structure -- is an essential ingredient of the accepted cosmological paradigm. The above drawback is in addition to the well known problem with classical instability against small perturbations of the background density originated from negativity of the sound speed squared.