Hartshorne's product conjecture for Chow groups of hypersurfaces

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, and let $h\in A^1(X)$ denote the hyperplane class. Hartshorne's product conjecture.

$$A^i(X)\cdot A^j(X)=\mathbb{Q}[h^{i+j}]\quad\text{for all }i,j>0\text{ such that }(i,j)\neq\left(\frac{n}{2},\frac{n}{2}\right).$$

This is presented as a consequence of Hartshorne's conjecture and the one-dimensionality of intersection with the hyperplane class; the source gives no resolution and notes that a more general complete-intersection version seems likely.