Do DESI-DR2 BAO data imply a coupling of dark matter and dark energy?
We revisit an interacting dark matter (DM) -- dark energy (DE) model characterized by the interaction function $Q = Γρ_x$, where $Γ$ is a constant coupling parameter and $ρ_x$ is the energy density of DE. This type of interaction is independent of the Hubble rate or other external parameters, but depends only on the fundamental properties of DE, such as its equation of state (EoS), $w_x$. We pay special attention to $w_x$ and study three distinct interacting scenarios distinguished by the nature of $w_x$, i.e. $w_x =-1$ (when DE corresponds to the vacuum energy), $w_{x} < -1$ (when DE has a phantom behavior), and $w_x > -1$ (quintessential DE). We constrain all of them using the most recent cosmological datasets, including CMB from Planck 2018, BAO from DESI DR2, and three compilations of SNIa (PantheonPlus, Union3, and DESY5). Our analyses reveal that evidence of interaction is supported in scenarios with $w_x =-1$ and $w_x < -1$ when all three datasets are combined, but from the Bayesian evidence analysis, $Λ$CDM remains favored over these interacting scenarios. Regarding the $S_8$ parameter, when $w_x > -1$, this interacting scenario leads to mildly lower estimates across all datasets.