The functorial determination conjecture for nef integral Cartier b-divisors

Let $\mathcal{C}$ be the category of pairs $(X,D)$ where $X$ is a reduced integral noetherian excellent $\mathbb{Q}$-scheme and $D$ is a nef integral Cartier b-divisor on $X$. A morphism $(Y,D')\to(X,D)$ consists of a morphism $f:Y\to X$ and an isomorphism $D'\cong f^*D$. Let $\mathcal{C}'$ be the subcategory with the same objects and only those morphisms whose underlying morphisms of schemes are regular, and let $\iota:\mathcal{C}'\to\mathcal{C}$ be the natural inclusion.

Functorial determination conjecture. There exist a covariant functor $Y:\mathcal{C}'\to\mathcal{C}$ and a natural transformation $F:Y\to\iota$ such that:

1. For each $(X,D)\in\mathcal{C}$, the morphism $F(X,D):Y(X,D)\to X$ is a modification that is a determination of $D$.
2. If $D\in\operatorname{CDiv}_{\mathbb{R}}X$, then $F(X,D)$ is an isomorphism.
3. For each morphism $f:(X',D')\to(X,D)$ in $\mathcal{C}$, the square

\xymatrix@C=50pt{ Y(X',D') \ar^{Y(f)}[r] \ar^{F(X',D')}[d] & Y(X,D) \ar^{F(X,D)}[d] \\ X' \ar^{f}[r] & X }

is cartesian in $\mathcal{C}$.

The conjecture is a functorial refinement of the existence of determinations for nef Cartier b-divisors, motivated by resolution of turning points. The supplied text does not state whether it has been resolved.