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 of [Bloch - Algebraic cycles and values of L-functions][1], though I’m particularly interested in the refined form of the conjecture, stated as conjecture 2.1.1 in [Bertolini, Darmon, and Prasanna - $p$-adic $L$-functions and the coniveau filtration on Chow groups][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