We search for a Gardner transition in glassy glycerol, a standard molecular glass, measuring the third harmonics cubic susceptibility χ_{3}^{(3)} from slightly below the usual glass transition temperature down to 10 K. According to the mean-field picture, if local motion within the glass were becoming highly correlated due to the emergence of a Gardner phase then χ_{3}^{(3)}, which is analogous to the dynamical spin-glass susceptibility, should increase and diverge at the Gardner transition temperature T_{G}. We find instead that upon cooling |χ_{3}^{(3)}| decreases by several orders of magnitude and becomes roughly constant in the regime 100-10 K. We rationalize our findings by assuming that the low temperature physics is described by localized excitations weakly interacting via a spin-glass dipolar pairwise interaction in a random magnetic field. Our quantitative estimations show that the spin-glass interaction is twenty to fifty times smaller than the local random field contribution, thus rationalizing the absence of the spin-glass Gardner phase. This hints at the fact that a Gardner phase may be suppressed in standard molecular glasses, but it also suggests ways to favor its existence in other amorphous solids and by changing the preparation protocol.