This is a follow-up to question Locally toric resolutions of compactifications, answered by Jason Starr.
In a series of papers (see https://arxiv.org/abs/math/9904076), Jaroslaw Wlodarczyk proves the "weak factorization theorem". Namely, let $f$ be a birational morphism of proper smooth varieties over a field of characteristic $0$, $$f:X \dashrightarrow Y,$$ which is an isomorphism on some open $U\subset X$ (to $U\subset Y$). Then $f$ can be factored as a sequence of morphisms $f_i:X_i\dashrightarrow X_{i+1}$ (with $X_0 = X, \,\, X_N = Y$ for some $N$) which are blow-ups or blow-downs at a smooth point. Moreover, this is true over any perfect field with resolution of singularities.
I want a local analogue. Namely, assume we have resolution of singularities in characteristic $p$ (as recently announced by Hironaka). Let $f:X\dashrightarrow Y$ be a birational map of regular varieties which fiber (with proper, though possibly singular fibers) over $\operatorname{Spec}\mathbb{Z}_p$ and such that $f$ is an isomorphism at the generic fiber. Can $f$ be written as a sequence of blow-ups and blow-downs? If not, can we impose some conditions on $X, Y$ so that it can?
