Kuznetsov's rationality conjecture for cubic fourfolds

Let $\mathcal{C}$ be the moduli space of cubic fourfolds, and let $\mathcal{C}_d$ denote a special divisor, consisting of cubic fourfolds with discriminant $d$. A discriminant $d$ is admissible when it satisfies the numerical conditions appearing in the classification of special cubic fourfolds associated with rationality. Kuznetsov's conjecture. A cubic fourfold $[X]\in\mathcal{C}$ is rational if and only if $[X]\in\mathcal{C}_d$, with $d$ admissible. This conjecture seeks to characterize rational cubic fourfolds through their membership in admissible special divisors; while many rational examples lie in such divisors, the asserted equivalence remains open in general.