On asymptotic stability on a center hypersurface at the soliton for even solutions of the NLKG when $2\ge p> \frac{5}{3}$

We extend the result M. Kowalczyk, Y. Martel, C. Muñoz, JEMS 2022, 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 $2\ge p> \frac{5}{3}$.