Let $X/K$ be a smooth projective variety of dimension $n$, and $\mathscr{X}/R$ be a dlt model of $X$ with reduced special fiber $\mathscr{X}_{k}=\sum_{\alpha}D_{\alpha}$, such that every $D_{\alpha}$ is a Cartier divisor.
Let $Z=D_{0}\cap D_{1}\cap\ldots\cap D_{n-r}$ be an $r$-dimensional stratum of $\mathscr{X}_{k}$, such that:

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  $\mathring{Z}\subset Z$ is a torus embedding, where $\mathring{Z}=Z\setminus\cup_{\alpha\neq 0,1,\ldots,n-r}D_{\alpha}$;

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  the conormal bundle $\nu_{Z/\mathscr{X}}^{*}$ is a nef vector bundle on $Z$;

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  for each $\alpha\notin\{0,...,n-r\}$, the intersection $D_{\alpha}\cap Z$ is either empty or connected.

Then the formal completion $\widehat{\mathscr{X}_{/Z}}$ is isomorphic to the formal completion of the normal bundle $\mathcal{N}=\nu_{Z/\mathscr{X}}$ along the zero section. In particular, $\mathscr{X}$ is toric along $Z$ (in the sense of Definition 1.2.6).