Do T asymmetries for neutrino oscillations in uniform matter have a CP-even component?

Observables of neutrino oscillations in matter have, in general, contributions from the effective matter potential. It contaminates the CP violation asymmetry adding a fake effect that has been recently disentangled from the genuine one by their different behavior under T and CPT. Is the genuine T-odd CPT-invariant component of the CP asymmetry coincident with the T asymmetry? Contrary to CP, matter effects in uniform matter cannot induce by themselves a non-vanishing T asymmetry; however, the question of the title remained open. We demonstrate that, in the presence of genuine CP violation, there is a new non-vanishing CP-even, and so CPT-odd, component in the T asymmetry in matter, which is of odd-parity in both the phase $\delta$ of the flavor mixing and the matter parameter $a$. The two disentangled components, genuine $A^\mathrm{T;CP}_{\alpha\beta}$ and fake $A^\mathrm{T;CPT}_{\alpha\beta}$, could be experimentally separated by the measurement of the two T asymmetries in matter ($\nu_\alpha \leftrightarrow \nu_\beta$) and ($\bar\nu_\alpha \leftrightarrow \bar\nu_\beta$). For the ($\nu_\mu \leftrightarrow \nu_e$) transitions, the energy dependence of the new $A^\mathrm{T;CPT}_{\mu e}$ component is like the matter-induced term $A^\mathrm{CP;CPT}_{\mu e}$ of the CP asymmetry which is odd under a change of the neutrino mass hierarchy. We have thus completed the physics involved in all observable asymmetries in matter by means of their disentanglement into the three independent components, genuine $A^\mathrm{CP;T}_{\alpha\beta}$ and fake $A^\mathrm{CP;CPT}_{\alpha\beta}$ and $A^\mathrm{T;CPT}_{\alpha\beta}$.