Possibility of reaching equilibrium starting with a nonequilibrium initial condition in the early radiation-domination

Update after @knzhou’s comment

If in a theory, the coupling of the dark matter (DM) field to the Standard Model (SM) fields is small enough, the rate of interaction of the DM particles in the primordial plasma of the early Universe may not be able to compete with the expansion rate of the Universe i.e. $$Gamma(T)<H(T)tag{1}$$ where $H(T)$ is the Hubble rate at temperature $T$ which varies as $H(T)sim T^2/M_{rm Pl}$ (assuming radiation domination). Therefore, it fails to enter thermal equilibrium.

Note that the both the interaction rate as well as the Hubble rate fall with the fall of temperature.

In order to reach thermal equilibrium at a later time, (and therefore, at a temperature $T'<T$), the inequality $(1)$ must be reversed i.e. $$Gamma(T’)> H(T’).tag{2}$$ must be satisfied at a later stage. This could happen if the interaction rate falls slower than the Hubble rate.

Does this really happen? How feasible it is that the inequality $(1)$ existing at a higher temperature $T$ will be flipped to become the inequality $(2)$ at a lower temperature $T’$?