Let $X/K$ be a smooth projective variety, and $\mathscr{X}/R$ be a dlt model of $X$ with reduced special fiber $\mathscr{X}_{k}$, such that the irreducible components of $\mathscr{X}_{k}$ are all Cartier divisors.
Let $Z$ be a stratum of $\mathscr{X}_{k}$, such that:

• •

  $\mathring{Z}\subset Z$ is a torus embedding, where $\mathring{Z}$ is the open stratum of $Z$;

• •

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

• •

  the intersection of $Z$ with any irreducible component of $\mathscr{X}_{k}$ is connected.

Then $\mathscr{X}$ is toric along $Z$ (in the sense of Definition 1.2.6).