(joint work with C. Berger and D.-C. Cisinski)
We show that the model category of n-braided operads in a k-truncated symmetric monoidal model category V is Quillen equivalent to the model category of (n+1)-braided operads if n>k+1. The Breen-Baez-Dolan stabilisation hypothesis follows from this theorem for any model of the theory of weak k-categories for which we know the existence of symmetric monoidal model structure. There are several such models in the literature and other models are expected to be, at least, equivalent to one of them.