On Asymptotic Stability on a Center Hypersurface at the Soliton for Even Solutions of the Nonlinear Klein–Gordon Equation When \(\boldsymbol{2 \ge p \gt \frac{5}{3}}\)

We extend the result of Kowalczyk, Martel, and Muñoz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133–2167] on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power nonlinear Klein–Gordon equation with , to the case . The result is attained performing new and refined estimates that allow us to close the argument for power law in the range .