What solves the Hubble tension in phenomenological dark energy models at background level?

Few phenomenological models tend to favour higher values of the Hubble parameter, often at the expense of invoking phantom transitions. These models achieve this without introducing additional parameters, akin to the simplicity of the concordance $\Lambda$CDM model. In this work, we investigate two such models -- Phenomenologically Emergent Dark Energy (PEDE) and Granda-Oliveros Holographic Dark Energy (GOHDE) -- to assess how correlations between $H_0$ and $\Omega_m$, as well as the choice of datasets, influence conclusions regarding their potential to address the Hubble tension at the background level. We find that minimally extended versions of these models favour notably low values for the Hubble parameter, with the perceived preference for higher values driven by the associated prior. Excluding BAO Ly$\alpha$-$H(z)$ data points at a redshift of $\sim 2.3$ results in a Hubble parameter that remains in significant tension with SH0ES measurements. In contrast, including these data points favours a higher $H_0$, as they suggest a relatively lower matter density within the framework of the assumed fiducial cosmology. Additionally, recent DESI DR1 and DR2 data exhibit mild tension with BAO-$H(z)$ estimates from SDSS. We demonstrate that the inclusion of stringent constraints, such as the CMB shift-parameter along with Pantheon$^+$, on the effective pressure less matter density significantly impacts the estimation of the Hubble parameter. Finally, reinterpreting these models in terms of interacting dark sectors with $Q = 3H\gamma_{\Lambda}\tilde{\rho}_{m}$ reveals that addressing the Hubble tension necessitates a varying $\gamma_{\Lambda}$ characterised by a singular sign-switch behaviour. This phantom behaviour, or equivalently, the onset of violation of the null energy condition in the future, is crucial for minimal models to solve the Hubble tension.