This follows from Corollary XIII.5 of Williamson's thesis Cylindrical model structures, where he proves that a morphism is a trivial cofibration if and only if it admits a strong deformation retraction. Since you already know $\gamma_X^0$ is a trivial cofibration, you're done.

You might also be interested in Section XII of Williamson's paper might also be of interest, as it provides conditions on the cylinder object in order that various associated maps are strong deformation retractions.

This follows from Corollary XIII.5 of Williamson's thesis Cylindrical model structures, where he proves that a morphism is a trivial cofibration (in his model structure determined by the cylinder object) if and only if it admits a strong deformation retraction. Since you already know $\gamma_X^0$ is a trivial cofibration, you're done.

You might also be interested in Section XII of Williamson's paper might also be of interest, as it provides conditions on the cylinder object in order that various associated maps are strong deformation retractions.