  We use the swap Monte Carlo algorithm to analyse the glassy behaviour of
sticky spheres in equilibrium conditions at densities where conventional
simulations and experiments fail to reach equilibrium, beyond predicted phase
transitions and dynamic singularities. We demonstrate the existence of a unique
ergodic region comprising all the distinct phases previously reported, except
for a phase-separated region at strong adhesion. All structural and dynamic
observables evolve gradually within this ergodic region, the physics evolving
smoothly from well-known hard sphere glassy behaviour at small adhesions and
large densities, to a more complex glassy regime characterised by
unusually-broad distributions of relaxation timescales and lengthscales at
large adhesions.
