Resolution of divisors by weakly admissible blow-ups

Let $(\Sigma_0,\Delta_0,\xi_0)$ be a coordinated normal crossing scheme over $k$, and let $\Phi_0$ be any effective divisor of $\Sigma_0$. A weakly admissible composition of blowing-ups over $(\Delta_0,\xi_0)$ is a morphism $\sigma:\Sigma\rightarrow\Sigma_0$ obtained by such a composition, and an extended pull-back is the coordinated normal crossing scheme $(\Sigma,\bar{\Delta},\bar{\xi})$ induced by $\sigma$. Resolution of divisors conjecture. There exist a weakly admissible composition of blowing-ups $\sigma:\Sigma\rightarrow\Sigma_0$ over $(\Delta_0,\xi_0)$ and an extended pull-back $(\Sigma,\bar{\Delta},\bar{\xi})$ of $(\Sigma_0,\Delta_0,\xi_0)$ by $\sigma$ satisfying

$$\operatorname{Supp}(\sigma^*(\Phi_0+\Delta_0))\subset \operatorname{Supp}(\bar{\Delta}).$$

This is a resolution statement asserting that the total transform of the chosen effective divisor together with the original normal-crossing boundary can be supported on the resulting boundary after a controlled sequence of blow-ups. The supplied text does not state whether this assertion is open, proved, or refuted.