Consider a finite-alphabet spin system evolving under a stochastic dynamics on a regular tree, starting from a Gibbs measure at some inverse temperature. As time progresses, the evolved measure may lose its Gibbs property, a phenomenon called the Gibbs-non-Gibbs transition. The open problem is whether the critical time at which this transition occurs is a continuous function of the initial temperature parameter, or whether there exist values of the initial temperature at which the critical transition time jumps discontinuously, potentially creating regimes where the system alternates between Gibbs and non-Gibbs behavior in a non-monotone fashion.