Previously it was shown by experiments that nanophase separation of incompatible main-and side-chain parts occurs in higher poly(n-alkyl acrylates) (PnAA) and poly(n-alkyl methacrylates) (PnAMA). Alkyl groups of different monomeric units of the polymer aggregate in the melt and form self-assembled alkyl nanodomains with a typical size of 0.5-2 nm. A polyethylene-like glass transition occurs within the alkyl nanodomains. We revisit the relaxation data and suggest that an unresolved Johari-Goldstein secondary relaxation is present in addition to the primary alpha-relaxation of this polyethylene-like glass transition. The ratio of the relaxation times of the two processes is consistent with the weakly cooperative nature of the alpha-relaxation, already indicated by its low steepness (fragility) index.