Multiplicative Chow–Künneth decomposition for Fano varieties of K3 type

Let $X$ be a smooth projective Fano variety of K3 type, meaning that $\dim X=2m$ and the Hodge numbers $h^{p,q}(X)$ vanish for all $p\not=q$, except that

$$h^{m-1,m+1}(X)=h^{m+1,m-1}(X)=1.$$

The multiplicative Chow–Künneth conjecture. The variety $X$ has a multiplicative Chow–Künneth decomposition. This is presented as a partial answer to the problem of describing varieties with such decompositions; the supplied text gives no resolution status.