**Abstract** : It is widely accepted that turbulent scalar flux can show the countergradient behavior almost everywhere within a premixed flame brush with the exception of a narrow zone at the leading edge (LE) of the flame where the flux always shows the gradient behavior. Moreover, many experts consider the existence of such a zone to be of crucial importance in order for the flame to be able to propagate into unburned mixture. The goal of the present work is to dispute this widely-recognized belief by studying an asymptotic case of density variations localized to infinitely thin, wrinkled flamelets that separate unburned and burned mixture and self-propagate at a finite speed into the former mixture. First, simple mathematical and physical examples are discussed in order to argue that a premixed flame can propagate into unburned mixture even if averaged scalar flux does not show the gradient behaviour at the LE. This phenomenon is associated with the straightforward influence of large-scale velocity oscillations (i.e. turbulent diffusion as far as a premixed turbulent flame is concerned) on the mean rate of product creation at the LE. Second, by considering a fully-developed, statistically stationary, planar, one-dimensional turbulent premixed flame, the following criterion is obtained. Turbulent scalar flux shows the countergradient behavior at the LE if turbulent burning velocity is less than the laminar flame speed multiplied by the density ratio and by a factor that (i) is equal to unity if perturbations of the local burning rate in flamelets are disregarded, but (ii) can substantially depend on the Lewis number and preferential diffusion effects if such perturbations are taken into account. Third, development of a premixed turbulent flame and straining of the flame brush by non-uniform mean flow are argued to suppress countergradient scalar flux at the LE.