The property that the pair $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$ is snc at a point of $\mathscr{X}_{k}$ only depends on the reduction of $\mathscr{X}$ modulo $t^{2}$. Thus by means of the approximation result in Proposition 4.3, we can reduce to the case where the model $\mathscr{X}$ is defined over a smooth algebraic $k$-curve; then the result follows from Theorem 4.5. ∎