  We investigate $p$-orbital Bose-Einstein condensates in both the square and
checkerboard lattice by numerically solving the Gross-Pitaevskii equation. The
periodic potential for the latter lattice is taken exactly from the recent
experiment [Nature Phys. 7, 147 (2011)]. It is confirmed that the staggered
orbital-current state is the lowest-energy state in the $p$ band. Our numerical
calculation further reveals that for both lattices the staggered $p$-orbital
state suffers Landau instability but the situation is remarkably different for
dynamical instability. A dynamically stable parameter region is found for the
checkerboard lattice, but not for the square.
