Is the following statement (maybe with more or less conditions) true?
Given a sequence of proper birational morphisms of integral varieties $X\leftarrow X_1\leftarrow X_2\leftarrow X_3\leftarrow... .$ Suppose there is another morphism $X\leftarrow Y$ which is finite birational with $Y$ integral and is factorized through every $X_i$. Then the sequence is finally stationary.
This question arised from remark 8.1.27 in Liu's Algebraic geometry and arithmetic curves, in which he said this is true for curves (if I understand it correctly).
-------------- edits: a case with two more assumptions --------------------
Denote the morphisms $X_i\rightarrow X_0:=X$ by $g_i$, the morphisms $Y\rightarrow X_i$ by $f_i$, and the morphisms $X_{i+1}\rightarrow X_i$ by $\pi_i$.
As $f_0$ is finite birational (hence surjective), so is every $f_i$, which means $g_i$ are quasi-finite and therefore also finite (By EGA-IV-8.11.1). So now all $f_i, g_i, \pi_i$ are finite birational (hence of finite fibres and surjective).
Now make the first more assumption that the integral varieties $X_i$ have ample $O_{X_i}$-modules (I think this assumption makes the above finte morphisms become projective in the defintition used in [Liu] and [Hartshorne], right?).
Then [Liu, 8.1.24] implies all $\pi_i$ are blowing-ups along some closed subscheme $Z_i$ of $X_i$. Now we make the second more assumption that $\pi_i$ maps $Z_{i+1}$ into $Z_i$ (set-theoretically) and $Z_0 (\subseteq X_0=X)$ is of dimension $0$.
Then the facts that all morphisms in consideration are surjective and of finite fibres and [Liu, 8.1.12(d)] imply for sufficiently large $n$, $\pi_n$ induces set-theoretically bijective maps $Z_{n+1}\rightarrow Z_n$, and (schematic) isomorphisms $X_{n+1}\setminus Z_{n+1}\rightarrow X_n\setminus Z_n$.
Finally I want use [Liu, 7.2.20(b)] to conclude this case, i.e. I want to show for every $x\in Z_{n+1}$ we have an isomophism of stalks $O_{X_n,\pi_n(x)}\rightarrow O_{X_{n+1},x}$, and I am stucked here.