Rational tower conjecture for unirational varieties

Let $X$ be a unirational variety over a field $k$. A rational tower is a sequence of dominant rational maps

$$X_n\longrightarrow X_{n-1}\longrightarrow\cdots\longrightarrow X_1\longrightarrow X_0:=X$$

over $k$, with rational source $X_n$ and geometrically rational irreducible generic fibers. Rational-tower conjecture. Then $X$ admits a rational tower. The conjecture would provide a systematic way to obtain unirational varieties from rational varieties through geometrically rational fibrations; its status is not resolved in the supplied text.