The Harris–Hassett rationality conjecture for Gushel–Mukai fourfolds

Let $X$ be a GM fourfold. An associated K3 surface of $X$ is a K3 surface arising from a period point of $X$ lying on a hypersurface $\mathcal{D}_d$, with a nef algebraic class $h$ satisfying $h^2=d$ and giving the corresponding polarized Hodge structure. The Harris–Hassett conjecture. The fourfold $X$ is rational if and only if it has an associated K3 surface.