The HEOM are an infinite system of equations that evolve individual auxiliary density operators. The equations are coupled via raising and lowering operators that allow the movement of phonons between ADOs. Once truncated, the resulting ADO structures can take on a variety of shapes with varying complexity. Within these structures ADOs are represented as elements lying along independent Matsubara axes, with their position along the axes being referred to as tiers. In this work, we show that ADOs contain virtual information whose flux through the hierarchy can be monitored. This can provide insight into the relative importance of individual ADOs in defining the bath. A qualitative understanding about the nature of the system-bath interaction can therefore be obtained directly from the virtual information metric BLP measure when it is applied to the ADOs. We have clearly demonstrated differing recurrence times of virtual information for ADOs across the hierarchy as a consequence of non-Markovian feedback. From this we can gain insight into the resulting impact on the spectral broadening and peak precision in 2DES. Finally, we employed a termination scheme based on the physical understanding afforded by this analysis. The resulting spectra from the secondary termination procedure are in qualitative agreement with those produced using the original hierarchy, thereby reducing computational cost for 2D spectra of OQS with complex baths. Future work will involve optimising the approach for the development of a new termination procedure systems with strong non-Markovian feedback.