Existence of birational sections over a center of an exceptional divisor Suppose $X$ is a variety with mild singularities (say terminal singularities), and $W$ is a center of some exceptional divisor over $X$ (i.e. there is a variety $Y$ with birational morphism $f: Y \to X$, and a divisor $E \subseteq Y$, such that $f(E) = W$). Suppose $W$ is normal (in my mind, $W$ is a minimal log canonical center). 
Question: Is there a resolution $g: \tilde X \to X$ which extracts the divisor $E$ and a subset $\tilde W \subset E$, such that the natural morphism $g|_{\tilde W}: \tilde W \to W$ is birational?
If the answer is no in general, is there any special case this holds? (I cannot assume that $X$ is smooth where the claim is certainly true.)
 A: The following argument should also work when $X$ has klt singularities.
By the result of Birkar-Cascini-Hacon-M$^{\rm c}$Kernan, we can find a birational model $f:X'\rightarrow X$ extracting only one divisor $E$ with $f(E)=W$. Since $-E$ is relative ample over $X$, there is an exact sequence,
$$f_*\mathcal{O}_{X'}=\mathcal{O}_X\rightarrow f_*\mathcal{O}_{E}\rightarrow R^1f_*\mathcal{O}_{X'}(-E)=0.$$
Since there is a factorization $$\mathcal{O}_X\rightarrow \mathcal{O}_W\rightarrow f_*\mathcal{O}_E,$$
the natural map $E\rightarrow W$ has connected fibers.
On the other hand, result of Hacon-M$^{\rm c}$Kernan asserts that $E\rightarrow W$ must have rationally connected fibers.
The only situation that I know where one can find a birational section is when $\dim W=1$: This follows from the result of Graber-Harris-Mazur-Starr. To get a resolution of $X$, one can simply replace $X'$ by a higher model. (The argument here is wrong since we only have RCC of fibers from H-M.)
However, if $\dim W\geq2$, it seems to me the answer is related to weak approximation problem as handled in the last paper. As I remember, this is a nontrivial problem.
When $W$ is also rationally connected, to have a rational section is still not an easy question. Artin-Mumford's conic bundle over $\mathbb{P}^2$ is a unirational but non-stably-rational variety.
Edit: As remark by Kostya, I made a mistake recalling the result of H-M, where it only guarantees RCC of fibers. From Mori's list of extremal contraction on smooth threefolds, one can contract a singular irreducible reduced quadric Q in P^3 to a terminal point. The surface Q is RCC, but not RC.
