Claborn–Fossum conjecture on Chow groups of regular local rings

Let $(A,\mathfrak{m}_A,L)$ be a regular local ring, and let $\operatorname{CH}^p(A)$ denote its codimension-$p$ Chow group. Claborn–Fossum Conjecture.

$$\operatorname{CH}^p(A)=0 \qquad\text{for all } p>0.$$

The conjecture extends the known vanishing of the divisor-class group of a regular local ring to algebraic cycles in every positive codimension. It has been settled in the equicharacteristic case by Panin and Quillen, and when $A$ is essentially smooth over a discrete valuation ring by Gillet–Levine.