Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an isomorphism on $U$, i.e. $X'\times_{X}U = U$, and write in this case shorthand $X'\to^UX$. Say a modification $X'\to^UX$ is a toric modification if $U, X, X'$ are all toric varieties with compatible action by the same $n$-dimensional torus.

**Definition.** We say that a modification $X'\to^U X$ is *of toric type* if for every $x\in X$, it looks like a toric modification locally. I.e., if for $X'_x$ the fiber of $X'$ over $x$, the map on formal neighborhoods $\hat{X}'_{x}\to \hat{x}$ coincides with an analogous formal neighborhood in for a toric modification such that $U\cap \hat{x}$ is torus-equivariant (it is probably better to say that the complement to $U$ is equivariant in $\hat{x}$).

**Definition.** Say that two compactifications $X, X'$ of $U$ are *toric-equivalent* if they are related by a chain of modifications of toric type.

For example, if $X$ is a surface with local coordinatex $x, y$ at a point $x_0$ and $U$ locally looks like the complement to the line $x = 0$ or the cross $xy = 0$, then the blow-up at $x_0$ of $X$ is a toric-type modification. Using just these local models, basic considerations about birational maps of surfaces (e.g. see Tony Pantev's answer to resolution of singularities on surfaces) imply that any two normal-crossings compactifications of a smooth surface are toric-equivalent.

My question: for what higher-dimensional varieties $U$ can we say that any two normal-crossings compactifications $X$ of $U$ are toric equivalent? Is there a nice class of comapctifications $X$ that exist for nice enough $U$ which are guaranteed to be toric-equivalent?