See the Supplementary Material for additional information on the hierarchical equations of motion (HEOM) as well as the mixed HEOM-Lindblad formalism; derivation for the inter-branch transition rate constants using Fermi's golden rule (FGR), where numerical illustrations are also attached; and the polariton lineshapes under a smaller cavity loss rate (Γ c = 8.83 meV), as well as the resulting motional narrowing characters.