Surjectivity conjecture for the generic-point cycle class map

Let $X/{\mathbb C}$ be a smooth projective variety, and let $r,m\geq 0$. Write $\lim(\textup{cl}_{r,m})$ for the direct-limit cycle class map at the generic point of $X$.

Generic-point cycle class conjecture. For all $r,m\geq 0$, statement (S3) holds. The supplied span does not reproduce statement (S3), so its precise mathematical content cannot be determined from the available context.