  The cosmological dark sector remains an enigma, offering numerous
possibilities for exploration. One particularly intriguing option is the
(non-minimal) interaction scenario between dark matter and dark energy. In this
paper, to investigate this scenario, we have implemented Binned and Gaussian
model-independent reconstructions for the interaction kernel alongside the
equation of state; while using data from BAOs, Pantheon+ and Cosmic
Chronometers. In addition to the reconstruction process, we conducted a model
selection to analyze how our methodology performed against the standard
$\Lambda$CDM model. The results revealed a slight indication, of at least
1$\sigma$ confidence level, for some oscillatory dynamics in the interaction
kernel and, as a by-product, also in the DE and DM. A consequence of this
outcome is the possibility of a sign change in the direction of the energy
transfer between DE and DM and a possible transition from a negative DE energy
density in early-times to a positive one at late-times. While our
reconstructions provided a better fit to the data compared to the standard
model, the Bayesian Evidence showed an intrinsic penalization due to the extra
degrees of freedom. Nevertheless these reconstructions could be used as a basis
for other physical models with lower complexity but similar behavior.
