Galkin–Shinder's rationality conjecture for cubic fourfolds

Let $X$ be a smooth cubic fourfold, let $F(X)$ be its Fano variety of lines, and let $S^{[2]}$ be the Hilbert scheme of length-two subschemes of a K3 surface $S$. Galkin–Shinder's conjecture. $X$ is rational if and only if $F(X)$ is birational to $S^{[2]}$ for some K3 surface $S$. The source presents this as an alternative rationality conjecture with stronger geometric evidence than the preceding two, but does not state a resolution.