  [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.
