Conjecture on representation of annealed time-ergodic invariant measures

Consider an annealed time-ergodic invariant measure for the SPDE discussed in the paper. A random field is weakly spatially stationary when it has the weak spatial stationarity property defined in the paper. Representation conjecture. Every annealed time-ergodic invariant measure is the law of a weakly spatially stationary random field. The paper presents this as an unproved conjecture, so whether all such invariant measures admit this representation remains open.