Hartshorne's Chow ring conjecture for smooth 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 conjecture.

$$A^i(X)=\mathbb{Q}[h^i]\quad\text{for all }i<\frac{n}{2}.$$

Apart from some easy results when $X$ has small degree and is Fano, the conjecture is completely open for $i\geq 2$ and appears highly challenging.