Numerical Chow-group vanishing conjecture for local complete intersections

Let $(A,m)$ be a $d$)-dimensional local domain with isolated singularity. Let $A_i(A)$ denote the $i$-th Chow group, let $G_0(A)$ be the Grothendieck group of finitely generated $A$-modules, and place a bar over a group to denote its quotient by numerical equivalence. Then:

Numerical Chow-group conjecture.

1. If $A$ is a complete intersection and $d$ is even, then

$$\overline{G_0(A)}_{\mathbb Q}=\mathbb Q[A],$$

equivalently, $\overline{A_i(A)}_{\mathbb Q}=0$ for $i<d$. 2. If $A$ is a complete intersection and $d$ is odd, then $\overline{A_i(A)}_{\mathbb Q}=0$ for $i\ne (d+1)/2,d$. 3. Without the complete-intersection assumption, if $i\leq d/2$, then $\overline{A_i(A)}_{\mathbb Q}=0$.

These assertions describe the expected numerical Chow groups of isolated-singularity local rings and explain the predicted vanishing of theta pairings. The paper proves related implications and establishes some cases, including the relevant complete-intersection statements in the geometric setting, but presents this collection as conjectural.