Uniqueness of Arrow-Debreu Equilibria

I have been trying to figure out conditions on the preferences of traders in an exchange economy such that the equilibrium prices are unique but I cannot be sure since all sources seem to be contradicting each other. On the Arrow-Debreu wikipedia page it says "In general, there may be many equilibria; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists." However, the Sonnenschein–Mantel–Debreu theorem does not tells us that the excess demand curve inherits the shape of the utility functions.

I do understand that for utilities satisfying the gross substitutes conditions the equilibrium prices are unique but are prices unique if we assume concave and twice differentiable utility functions?