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In natural and synthetic excitonic networks such as photosynthetic complexes and solar cells, we are faced with non-unitary quantum evolution due to dissipative and decoherent interaction with the environment. Studies have shown that dissipative quantum evolution surpasses both classical and purely quantum transport (for interesting recent examples see[Whitfield10,SMPE12]). A widely studied process of such dissipative exciton transport is the one occurring in the Fenna-Matthews-Olsen complex (FMO), which connects the photosynthetic antenna to a reaction centre in green sulphur bacteria[MRLA08,caruso09,fleming10,ringsmuth2012]. Due to the low light exposure of these bacteria, there is evolutionary pressure to optimize exciton transport. Therefore, the site energies and site-to-site couplings in the system are evolutionarily optimized, yielding a highly efficient transport[lloyd2011]. However, it is an open question whether or not there occurs time-reversal asymmetric hoping terms in these systems, and whether these are optimized. Recent 2D Electronic Spectroscopy results lead to the conclusion that , e.g., in the light harvesting complex LH2 hopping terms with complex phases are indeed present [Engel].