The finiteness conjecture for radical mixed difference ideals

Let $k$ be a $\sigma$-field and $R$ a finitely $\sigma$-generated $k$-$\sigma$-algebra. A $\sigma$-ideal is radical if it is radical as an ordinary ideal, and mixed if, whenever $ab\in I$, one has $a\sigma(b)\in I$. Hrushovski's finiteness conjecture. Every ascending chain of radical, mixed $\sigma$-ideals in $R$ is finite. This would strengthen the classical Ritt--Raudenbusch basis theorem for difference algebras; it is posed as a question raised by E. Hrushovski, and its resolution is not given in the supplied text.