Vishik's conjecture on local Chow groups and numerical equivalence

Let $k$ be a flexible field. The local Chow groups are the groups obtained from the local Chow theory over $k$, while $\operatorname{Ch}^*_{Num}$ denotes Chow groups with $\mathbb{F}_p$-coefficients modulo numerical equivalence. Vishik's conjecture. Over a flexible field, these two cohomology theories coincide:

$$\operatorname{Ch}^*_{k/k}=\operatorname{Ch}^*_{Num}.$$

The conjecture would identify isotropic or local Chow motives with numerical Chow motives. The paper establishes a natural surjection from the isotropic Chow groups to the numerical ones, but does not prove that this surjection is an isomorphism over flexible fields.