To further gauge the consequence of the timedependent rate from NE-FGR theory, the E-FGR population dynamics are compared against that of NE-FGR as well as a time-nonlocal (TNL) master equation, which is a further generalization of NE-FGR. In our model system, given the relatively weak system-bath coupling λ < k B T chosen for the present trial, starting from the upper polariton yields coinciding dynamics for E-FGR, NE-FGR, and TNL. Hence, the non-Markovian effects are not significant for population dynamics starting from the |+⟩ state for the current model system investigated in this work, but our NE-FGR formalism is general for the future investigation to explore any significant difference for other HTC model systems with different J(ω). Further comparing these dynamics to the numerically exact HEOM simulations for the case N = 5, we find that E-FGR, NE-FGR, and TNL all give very accurate dynamics in this weak system-bath coupling regime. Since the bath further decouples from the system (note the system-bath coupling ∝ c a / √ N in Eq. A2) under the collective limit (N → ∞), TNL and NE-FGR are expected to be accurate in the experimentally relevant setting as well because the second order perturbation approximation is valid.