  The tunneling dynamics of a magnetic $^{164}$Dy quantum gas in an elongated
or pancake skewed double-well trap is investigated with a time-dependent
extended Gross-Pitaevskii approach. Upon lifting the energy offset, different
tunneling regimes can be identified. In the elongated trap and for sufficiently
large offset, the different configurations exhibit collective macroscopic
tunneling. For smaller offset, partial reflection from and transmission through
the barrier lead to density accumulation in both wells, and eventually to
tunneling-locking. One can also reach the macroscopic self-trapping regime for
increasing relative dipolar interaction strength, while tunneling vanishes for
large barrier heights. A richer dynamical behavior is observed for the
pancake-like trap. For instance, the supersolid maintains its shape, while the
superfluid density gets distorted signifying the emergence of peculiar
excitation patterns in the macroscopic tunneling regime. The findings reported
here may offer new ways to probe distinctive dynamical features in the
supersolid and droplet regimes.
