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A dynamical resolution to the cosmological constant fine-tuning problem has been previously put forward, based on a scalar-tensor gravitational theory possessing de Sitter attractor solutions characterized by a small Hubble expansion rate, irrespective of an initially large vacuum energy. We show that a technically natural subregion of the parameter space yields a cosmological evolution through radiation- and matter-dominated eras that is essentially indistinguishable from that predicted by General Relativity. Similarly, the proposed model automatically satisfies the observational constraints on a fifth force mediated by the new scalar degree of freedom.