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

.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 \(p> 3\) , to the case \(2≥ p> 5/3\) . The result is attained performing new and refined estimates that allow us to close the argument for power law in the range \(2≥ p> 5/3\) .