We consider a point in ${\mathscr{X}}^{\text{sing}}\cap D_{1}\cap D_{2}\cap D_{3}$; the singular points in the other strata curves can be treated analogously. Étale locally around such a point, $\mathscr{X}$ is isomorphic to the toric variety $\mathscr{U}\coloneqq V(xyz-wt)\subset\mathbb{A}^{5}_{k}$, where

$$D_{1}|_{\mathscr{U}}=\{x=t=0\},\quad D_{2}|_{\mathscr{U}}=\{y=t=0\}\,\text{ and }\,D_{3}|_{\mathscr{U}}=\{z=t=0\}$$

are the components of the special fiber $\{t=0\}$ in $\mathscr{U}$. We still denote these by $D_{1},D_{2}$ and $D_{3}$; they form the toric boundary of $\mathscr{U}$ together with

$$D_{1}^{\prime}|_{\mathscr{U}}=\{x=w=0\},\quad D_{2}^{\prime}|_{\mathscr{U}}=\{y=w=0\}\,\text{ and }\,D_{3}^{\prime}|_{\mathscr{U}}=\{z=w=0\}.$$

The pair $(\mathscr{X},\mathscr{X}_{k})_{|\mathscr{U}}=(\mathscr{U},\sum_{i=1,2,3}D_{1}|_{\mathscr{U}})$ is log canonical by [CLS11, Proposition 11.4.24].

We denote the strata surfaces of the special fiber by $D_{ij}\coloneqq D_{j}\cap D_{j}$, for $i,j\in\{1,2,3\}$ and $i\neq j$, and the stratum curve by $D_{123}\coloneqq\{x=y=z=t=0\}$. The singular locus ${\mathscr{U}}^{\text{sing}}$ consists of the torus invariant curves

$$C_{12}|_{\mathscr{U}}=\{x=y=w=t=0\}=D_{1}\cap D_{2}\cap D_{1}^{\prime}\cap D_{2}^{\prime},$$

$$C_{13}|_{\mathscr{U}}=\{x=z=t=w=0\}\,\text{ and }\,C_{23}|_{\mathscr{U}}=\{y=z=w=t=0\},$$

which intersect each other at the torus invariant point $p\coloneqq\{x=y=z=w=t=0\}$.

The toric blow-up $G_{1}:\mathscr{U}_{1}\coloneqq\Bl_{D_{1}}\mathscr{U}\rightarrow\mathscr{U}$ along $D_{1}$ resolves the singularities along $C_{12}$ and $C_{13}$ except at the point $p$. The exceptional locus of $G_{1}$ consists of two surfaces $S_{12}$ and $S_{13}$ intersecting each other along a curve: these surfaces are mapped by $G_{1}$ to the respective singular curves, are contained in the strict transform of $D_{1}$, and correspond to two new edges in the slice of the fan of $\mathscr{U}_{1}$. The intersection of $S_{12}$ and $S_{13}$ corresponds to a new $2$-dimensional face in the slice of the fan. With a slight abuse of notation we keep the same notation for the strict transforms in $\mathscr{U}_{1}$.

A resolution of $\mathscr{U}$ is given by the composition of $G_{1}$ and the toric blow-up $G_{12}:\mathscr{U}_{12}\coloneqq\Bl_{D_{2}}\mathscr{U}_{1}\rightarrow\mathscr{U}_{1}$ along $D_{2}$; the latter indeed resolves the singularities along $C_{23}$. The exceptional locus of $G_{12}$ is a surface $S_{23}$, which is mapped by $G_{12}$ to $C_{23}$ and is contained in the strict transform of $D_{2}$. The morphism $G_{12}$ induces a new $2$-dimensional face in the slice of the fan of $\mathscr{U}_{12}$, and a new edge corresponding to the surface $S_{23}$. In particular, after the blow-up $G_{12}$, the strict transforms of the surface $S_{12}$ and of the divisor $D_{3}$ have empty intersection.

The small resolution $\mathscr{U}_{12}$ of $\mathscr{U}$ induces an isomorphism on the strict transform of $D_{13}$ and of $D_{23}$, while its restriction to the strict transform of $D_{12}$ is the blow-up of $D_{12}$ along a general point. These facts can be checked computing the charts of the blow-ups $G_{1}$ and $G_{12}$. Alternatively, they can be verified looking at the fans of the strata surfaces $D_{ij}$ in the slice of the fans of $\mathscr{U}$, $\mathscr{U}_{1}$ and $\mathscr{U}_{12}$; indeed, the fan of $D_{ij}$ is induced by the intersection of the slice with a normal plane to the edge corresponding to $D_{ij}$.