While no physical interpretation can be garnered from information flow between the auxiliaries, or the Matsubara dimensions which they inhabit, the specific structure of the Matsubara space containing the ADOs nevertheless directly impacts the nature of the system-bath coupling and therefore dynamics of the specific OQS. The nature of the auxiliaries, and their intrinsic relation to the system parameters, are discussed within Fay et al. , and Yan et al. , but the level of physical insight that can be gained from the auxiliaries is not discussed. In contrast, authors such as Zhu et al. propose an explicit system-bath correlation by creating a collective bath coordinate from the auxiliaries. Additionally Xing et al. produce an imaginary time HEOM and use this to explicitly calculate real correlation functions through the path integral formalism. Both of these methods are impactful studies of the auxiliary density operators, but both craft bespoke operations to analyse the auxiliaries: neither consider the transfer of information between ADOs and how this might be used. In this study we consider a vibronic molecule with a fundamental vibration subsumed into the spectral density to systematically analyse how this impacts information flow within the ADOs. This is linked directly to electronic spectral broadening. From this we aim to ascertain the form and magnitude of information flow through the ADOs. We term this information, virtual information in order to abstract it from the physical quantum information within the density matrix; the former being a mathematical tool, and the latter an experimentally observable quantity. We analyse and characterise behaviours within the ADOs with respect to their Matsubara coordinate via the BLP metric. From this we draw conclusions to optimise computational efficiency for OQS dynamics in general and apply this specifically to two-dimensional electronic spectroscopy (2DES) simulations. We propose a new hierarchy termination constraint based on the BLP metric which is useful when modelling vibronic molecules within the condensed phase.