Embedded resolution with an snc boundary

Let $\bar{X}$ be a smooth scheme over a perfect field $k$, let $D\subset\bar{X}$ be an snc divisor, and let $\bar{Z}\subset\bar{X}$ be a reduced closed subscheme that is smooth over $\bar{X}\setminus D$, with $\bar{Z}\setminus D$ smooth over $k$.

Embedded resolution with boundary. The triple $(\bar{X},D,\bar{Z})$ admits a resolution by blowups in smooth centers

$$\bar{Y}:=\bar{X}_m\to\bar{X}_{m-1}\to\dots\to\bar{X}_2\to\bar{X}_1:=\bar{X},$$

such that each $\bar{X}_{i+1}\xrightarrow{\pi_i}\bar{X}_i$ is a blowup in a smooth center $W_i$ with exceptional divisor $E_{i+1}$; with $D_1:=D$ and $D_{i+1}$ the strict transform of $D_i$ in $\bar{X}_{i+1}$, each $D_i$ is an snc divisor and $W_i\subset D_i$; and $\pi^{-1}(\bar{Z})$ is a smooth $k$-scheme.

This stronger boundary-compatible resolution conjecture implies embedded resolution over a perfect field. The source does not state a general resolution result for it; only the related lower-dimensional embedded-resolution results are given.