Finite-product independence conjecture for measure-preserving dynamical systems

Let $\mathcal{F}$ be a family of subsets of the positive integers, and let a measure-preserving dynamical system (m.d.s.) be called $\mathcal{F}$-independent when it has the independence property associated with $\mathcal{F}$. Given finitely many $\mathcal{F}$-independent m.d.s., form their product system. Finite-product independence conjecture. For any family $\mathcal{F}$, the product of finitely many $\mathcal{F}$-independent m.d.s. remains $\mathcal{F}$-independent. The preceding cases establish this for several important families, including weak mixing, completely positive entropy, and the families denoted by $\mathcal{F}_{\inf}$, $\mathcal{F}_{\mathrm{ip}}$, and $\mathcal{F}_{\mathrm{pd}}$; the assertion for an arbitrary family is left as a conjecture.