Hassett–Kuznetsov rationality conjecture for cubic fourfolds

Let $X$ be a smooth cubic fourfold, and let $\mathcal{C}_d$ denote the Hassett divisor parametrizing special cubic fourfolds with a labelling of discriminant $d$. The numerical condition is

$$d>6$$

and $d$ is not divisible by $4$, $9$, or any prime $p\equiv 2\pmod 3$. Hassett–Kuznetsov rationality conjecture. The fourfold $X$ is rational if and only if $X\in\mathcal{C}_d$ for some $d$ satisfying this numerical condition. This is presented in the source as a conjecture, with no resolution supplied there.