The conjecture on exponentially long-lived discrete time-crystalline behavior

Consider the Floquet system defined by the Hamiltonian model in equation, with parameters $(\mathbf h,\mathbf J)\in[-20,20]^{\times(3N-1)}$. Discrete time-crystal stability conjecture. There is $(\mathbf h,\mathbf J)\in[-20,20]^{\times(3N-1)}$ at which discrete time-crystalline behavior is stable up to time $e^{\Omega(N)}$ with respect to extensive periodic perturbations of period $1$. This conjecture concerns stability at short and intermediate times, whereas the paper explains that discrete time-crystalline behavior at the longest time scales is unstable under infinitesimal symmetry-preserving perturbations. The source presents proving or disproving this statement as an important question, although the supplied parser status marks it as disproved.