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, 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. If so, what evidence exists for the conjecture?