  We show that the sublinear Morse boundary of every CAT(0) space continuously
injects into the Gromov boundary of a hyperbolic space, which was not
previously known even for all CAT(0) cube complexes. Our work utilizes the
curtain machinery introduced by Petyt-Spriano-Zalloum. Curtains are more
general combinatorial analogues of hyperplanes in cube complexes, and we
develop multiple curtain characterizations of the sublinear Morse property
along the way. Our results answer multiple questions of Petyt-Spriano-Zalloum.
