  We prove that the sublinearly Morse boundary of every known cubulated group
continuously injects in the Gromov boundary of a certain hyperbolic graph. We
also show that for all CAT(0) cube complexes, convergence to sublinearly Morse
geodesic rays has a simple combinatorial description using the hyperplanes
crossed by such sequences. As an application of this combinatorial description,
we show that a certain subspace of the Roller boundary continously surjects on
the subspace of the visual boundary consisting of sublinearly Morse geodesic
rays.
