The dominant tower conjecture for projective varieties

Let $k$ be a field and let $n$ be a positive integer. A smooth projective variety $X$ of dimension $n$ is endowed with a tower of smooth fibrations

$$X\to X_1\to\cdots\to X_n$$

where $\dim X_i=n-i$ for each $i$. Dominant tower conjecture. For any field $k$ and positive integer $n$, the class of such $n$-dimensional smooth projective varieties is dominant: for every projective $n$-dimensional variety $V$ over $k$, there exists a member $X$ of the class and a surjective $k$-morphism $X\to V$. This conjecture would provide a uniformization-type strengthening of de Jong's alteration theorem, but the general assertion remains open, including related questions about minimal dominant classes.