Is there a version of the Bloch-Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 [here][1], thought I’m particularly interested in the refined form of the conjecture, stated as conjecture 2.1.1 [here][2]. If so, what evidence exists for the conjecture?

  [1]: https://eudml.org/doc/152633
  [2]: https://dept.math.lsa.umich.edu/~kartikp/research/BDP5-hl.pdf#page7