  We investigate the non-equilibrium behavior of a fully-connected (or
all-to-all coupled) Bose-Hubbard model after a Mott to superfluid quench, in
the limit of large boson densities and for an arbitrary number $V$ of lattice
sites, with potential relevance in experiments ranging from cold atoms to
superconducting qubits. By means of the truncated Wigner approximation, we
predict that crossing a critical quench strength the system undergoes a
dynamical phase transition between two regimes that are characterized at long
times either by an inhomogeneous population of the lattice (i.e. macroscopical
self-trapping) or by the tendency of the mean-field bosonic variables to split
into two groups with phase difference $\pi$, that we refer to as
$\pi$-synchronization. We show the latter process to be intimately connected to
the presence, only for $V \ge 4$, of a manifold of infinitely many fixed points
of the dynamical equations. Finally, we show that no fine-tuning of the model
parameters is needed for the emergence of such $\pi$-synchronization, that is
in fact found to vanish smoothly in presence of an increasing site-dependent
disorder, in what we call a synchronization crossover.
