  In the effort to define the main features a successful solution to the Hubble
tension should have, growing evidence has emerged pointing to the need for
models able to modify the expansion (and possibly thermal) history of the
universe both prior and after recombination, and that could thereby restore the
overall concordance between early- and late-time observations without
introducing nor worsening other cosmological tensions in the process. In
precisely this spirit, here we consider a multi-interacting dark energy model
with two complementary interaction channels: one with the dark matter and one
with the photons. The former most significantly affects the matter dominated
epoch as well as the very late universe, and has been shown to be able to
significantly resolve the $S_8$ tension. The latter has been introduced in this
work to allow for a larger and natural time dependence of the evolution of the
universe since it mostly impacts the radiation dominated epoch as well as the
temperature scaling of the photons, as extensively explained in the text. As a
result, considering data from Planck+BAO+Pantheon+KV450+DES (which can be
combined since the $S_8$ value is lowered even neglecting the weak lensing
data) we find that the significance of the $H_0$ tension only reduces to about
3.5$\sigma$, while that of the $S_8$ tension falls below the 2$\sigma$ level.
Also, the statistical analysis we perform strongly favors $\Lambda$CDM in all
considered cases. Overall, we conclude that the specific multi-interacting dark
energy model considered here, despite its broad generality and very rich
cosmological phenomenology, cannot successfully restore the aforementioned
overall concordance between early- and late-time observations.
