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Two-dimensional optical spectroscopy experiments have shown that exciton transfer pathways in the Fenna-Matthews-Olson (FMO) photosynthetic complex differ drastically under reduced and oxidised conditions, suggesting a functional role for collective vibronic mechanisms that may be active in the reduced form but attenuated in the oxidised state. Higgins et al. [PNAS 118 (11) e2018240118 (2021)] used Redfield theory to link the experimental observations to altered exciton transfer rates due to oxidative onsite energy shifts that detune excitonic energy gaps from a specific vibrational frequency of the bacteriochlorophyll (BChl) a. Using a memory kernel formulation of the hierarchical equations of motion, we present non-perturbative estimations of transfer rates that yield a modified physical picture. Our findings indicate that onsite energy shifts alone cannot reproduce the observed rate changes in oxidative environments, either qualitatively or quantitatively. By systematically examining combined changes both in site energies and the local environment for the oxidised complex, while maintaining consistency with absorption spectra, our results suggest that vibronic tuning of transfer rates may indeed be active in the reduced complex. However, we achieve qualitative, but not quantitative, agreement with the experimentally measured rates. Our analysis indicates potential limitations of the FMO electronic Hamiltonian, which was originally derived by fitting spectra to second-order cumulant and Redfield theories. This suggests that reassessment of these electronic parameters with a non-perturbative scheme, or derived from first principles, is essential for a consistent and accurate understanding of exciton dynamics in FMO under varying redox conditions., Comment: 14 pages, 7 figures