  Horndeski models with a de Sitter critical point for any kind of material
content may provide a mechanism to alleviate the cosmological constant problem.
Moreover, they could allow us to understand the current accelerated expansion
of the universe as the result of the dynamical approach to the critical point
when it is an attractor. We show that this critical point is indeed an
attractor for the shift- symmetric subfamily of models with these
characteristics. We study the cosmological scenario that results when
considering radiation and matter content, and conclude that their background
dynamics is compatible with the latest observational data.
