It is shown in [NX16a, 4.2.4] that $\mathrm{Sk}(X)$ is a strong deformation retract of $X^{\mathrm{an}}$. This implies that every continuous retraction $X^{\mathrm{an}}\to\mathrm{Sk}(X)$ is homotopic to the identity on $X^{\mathrm{an}}$ relative to $\mathrm{Sk}(X)$; in particular, this is true for the retraction $\rho_{\mathscr{X}}$. ∎