Triple-intersection conjecture for Chow groups of smooth hypersurfaces

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, let $h\in A^1(X)$ denote the hyperplane class, and let $i_1,i_2,i_3>0$. Triple-intersection conjecture.

$$A^{i_1}(X)\cdot A^{i_2}(X)\cdot A^{i_3}(X)=\mathbb{Q}[h^{i_1+i_2+i_3}].$$

The source presents this as a consequence of Hartshorne's conjecture in the discussion of general-type hypersurfaces. No resolution is given.