Equivalent minimal-prime finiteness conjecture for difference algebras

Let $k$ be a $\sigma$-field and $R$ a finitely $\sigma$-generated $k$-$\sigma$-algebra. A prime

$$-ideal$$

of $R$ is minimal if every prime

$$-ideal$$

with $\subset$ satisfies $=$. Minimal-prime finiteness conjecture. The set of minimal prime

-ideals of $R$ is finite. By the cited result of Hrushovski, this is equivalent to the finiteness conjecture for ascending chains of radical, mixed

-ideals. Its general status is not resolved in the supplied source.