The rationality conjecture for Gushel–Mukai fourfolds

Let $X$ be a rational Gushel–Mukai fourfold, and let $\mathcal{A}_X$ denote its GM category. A rationality conjecture. Then

$$\mathcal{A}_X\simeq\operatorname{D^b}(S)$$

for some K3 surface $S$. This is the GM analogue of the rationality conjecture for cubic fourfolds. The conjecture is verified for a certain family of rational GM fourfolds, while the GM category of a very general GM fourfold is not geometric; the general implication remains open.