By the definition of a dlt-pair, the scheme $\mathscr{X}$ is regular at the generic point of every stratum of $D$, and at all the other points $x\in\mathscr{X}$, the minimal log discrepancy $\mathrm{mld}_{x}(\mathscr{X},D)$ is positive. Since $K_{\mathscr{X}/R}+D$ is Cartier, $\mathrm{mld}_{x}(\mathscr{X},D)$ is an integer, and therefore at least $1$. The inequality

$$\mathrm{mld}_{x}(\mathscr{X},0)>\mathrm{mld}_{x}(\mathscr{X},D)\geq 1$$

now implies that $\mathscr{X}$ is terminal. In particular, it is regular in codimension two. ∎