[Abridged] A new scenario --early disc accretion-- has been proposed very recently to explain the origin of the multiple population phenomenon in Galactic globular clusters. It envisages the possibility that a fraction of low- and very low-mass cluster stars may accrete the ejecta of interacting massive binary (and possibly also fast rotating massive) stars during the fully convective, pre-main sequence stage, to reproduce the CN and ONa anticorrelations observed among stars in individual clusters. This scenario is assumed to be able to explain the presence (and properties) of the multiple populations in the majority of globular clusters in the Milky Way. Here we have considered the well studied cluster NGC 2808, which displays a triple main sequence with well defined and separate He abundances. Knowledge of these abundances allowed us to put strong constraints on the He mass fraction and amount of matter to be accreted by low-mass pre-main sequence stars. We find that the minimum He mass fraction in the accreted gas has to be $sim0.44$ to produce the observed sequences and that at fixed initial mass of the accreting star, different efficiencies for the accretion are required to produce stars placed onto the multiple main sequences. This may be explained by differences in the orbital properties of the progenitors and/or different spatial distribution of intracluster gas with varying He abundances. Both O-Na and C-N anticorrelations appear naturally along the main sequences, once considering the predicted relationship between He and CNONa abundances in the ejecta of the polluters. As a consequence of the accretion, we predict no discontinuity between the abundance ranges covered by intermediate and blue main sequence stars, but we find a sizeable (several 0.1 dex) discontinuity of the N and Na abundances between objects on the intermediate and red main sequences.
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