Two nonisomorphic flows with the same very weak Bernoulli partition on a base (Q1076834)

It is known that there are uncountable many nonisomorphic flows built under step functions with the same Bernoulli base. It turns out that there is not a good characterization of the flows built with a very weak Bernoulli base partition, too. Namely, as the author has shown, there exists two nonisomorphic (even after time scaling) flows with the same finite very weak Bernoulli partition on a base under two step functions, each having the property that its values are independent over the rationals. It is an open problem to construct infinitely many nonisomorphic flows with the above properties. The method used by the autor (following an argument of D. J. Rudolph) fails to solve it.