Weak Lefschetz conjecture for Chow groups

Let $X/k$ be a smooth projective variety over a field $k$ of characteristic $0$, and let $L$ be an ample divisor. The natural restriction map is

$$CH^p(X)\longrightarrow CH^p(L).$$

Weak Lefschetz conjecture. This map is an isomorphism when $2p<\dim(L)$ and is injective modulo torsion when $2p=\dim(L)$. This is the Chow-group analogue of the classical Weak Lefschetz theorem. The paper studies its infinitesimal part using Milnor $K$-theory and Bloch's formula; the full conjecture is not resolved here.