  A giant planet has been recently resolved at a projected distance of 730 au
from the tight pair of young ($\sim$ 13 Myr) intermediate-mass stars HD
106906AB in the Lower Centaurus Crux (LCC) group. The stars are surrounded by a
debris disk which displays a ring-like morphology and strong asymmetries at
multiple scales. We aim at studying the likelihood of a scenario where the
planet formed closer to the stars in the disk, underwent inward disk-induced
migration, and got scattered away by the binary star before being stabilized by
a close encounter (fly-by). We performed semi-analytical calculations and
numerical simulations (Swift_HJS package) to model the interactions between the
planet and the two stars. We accounted for the migration as a simple force. We
studied the LCC kinematics to set constraints on the local density of stars,
and therefore on the fly-by likelihood. We performed N-body simulations to
determine the effects of the planet trajectories (ejection and secular effects)
onto the disk morphology. The combination of the migration and mean-motion
resonances with the binary star (often 1:6) can eject the planet. Nonetheless,
we estimate that the fly-by hypothesis decreases the scenario probability to
less than $10^{-7}$ for a derived local density of stars of 0.11
stars/pc$^{3}$. We show that the concomitant effect of the planet and stars
trajectories induce spiral-features in the disk which may correspond to the
observed asymmetries. Moreover, the present disk shape suggests that the planet
is on an eccentric orbit. The scenario we explored is a natural hypothesis if
the planet formed within a disk. Conversely, its low probability of occurrence
and the fact that HD 106906 b shares some characteristics with other systems in
Sco-Cen (e.g. HIP 78530, in terms of mass ratio and separation) may indicate an
alternative formation pathway for those objects.
