  We prove uniform synchronisation by noise with rates for the stochastic
quantisation equation in dimensions two and three. The proof relies on a
combination of coming down from infinity estimates and the framework of
order-preserving Markov semigroups derived in [Butkovsky, Scheutzow; 2019]. In
particular, it is shown that this framework can be applied to the case of state
spaces given in terms of H\"older spaces of negative exponent.
