The MCK decomposition conjecture for Fano varieties of K3 type

Let $X$) be a Fano variety of K3 type, meaning that $X$ has dimension $2d$ and its Hodge numbers satisfy

$$h^{p,q}(X)=0$$

for all $p\not=q$, except that $h^{d-1,d+1}(X)=h^{d+1,d-1}(X)=1$. Here, an MCK decomposition is a multiplicative Chow–Künneth decomposition of the motive of a variety.

MCK decomposition conjecture. Every Fano variety $X$ of K3 type has an MCK decomposition.

The conjecture is motivated by the expected relationship between Fano varieties of K3 type and hyperkähler varieties, which are themselves expected to admit MCK decompositions. Its resolution is not stated in the supplied text, so its status remains open.