The canonical scalar-tensor theory model which exhibits spontaneous scalarization in the strong-gravity regime of neutron stars has long been known to predict a cosmological evolution for the scalar field which generically results in severe violations of present-day Solar System constraints on deviations from general relativity. We study if this tension can be alleviated by generalizing this model to include a disformal coupling between the scalar field φ and matter, where the Jordan frame

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... c g̃_μν is related to the Einstein frame one g_μν by g̃_μν=A(φ)^2 (g_μν+Λ ∂_μφ ∂_νφ). We find that this broader theory admits a late-time attractor mechanism towards general relativity. However, the existence of this attractor requires a value of disformal scale of the order Λ≳ H_0^-2, where H_0 is the Hubble parameter of today, which is much larger than the scale relevant for spontaneous scalarization of neutron stars Λ∼ R_s^2 with R_s (∼ 10^-22 H_0^-1) being the typical radius of these stars. The large values of Λ necessary for the attractor mechanism (i) suppress spontaneous scalarization altogether inside neutron stars and (ii) induce ghost instabilities on scalar field fluctuations, thus preventing a resolution of the tension. We argue that the problem arises because our disformal coupling involves a dimensionful parameter.