  We show that the limiting Khovanov chain complex of any infinite positive
braid categorifies the Jones-Wenzl projector. This result extends Lev
Rozansky's categorification of the Jones-Wenzl projectors using the limiting
complex of infinite torus braids. We also show a similar result for the
limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such
braids. Extensions to more general infinite braids are also considered.
