The B-strong desingularization conjecture

A $B$-scheme is a scheme $X$ equipped with a boundary $B$; here $(X,B)$ is assumed to be qe and generically reduced, and the scheme has characteristic zero. Write ${\mathcal F}(X,B)$ for a $B$-strong desingularization, and call a morphism exact regular when it is an exact regular morphism of $B$-schemes. The $B$-strong desingularization conjecture. For any qe generically reduced $B$-scheme $(X,B)$ of characteristic zero there exists a $B$-strong desingularization ${\mathcal F}(X,B)$ which is functorial in all exact regular morphisms. The conjecture is intended to give a common formulation of embedded and non-embedded desingularization, although it does not include principalization. The source states that it should be provable in characteristic zero with current techniques, but does not give a proof here.