The rationality conjecture for cubic fourfolds with associated K3 surfaces

Let $X$ be a smooth cubic fourfold in $\mathbf{P}^5$. Let $A(X)=H^4(X,\mathbf{Z})\cap H^{2,2}(X)$, and let $\mathcal{C}_d$ denote the Hassett divisor of special cubic fourfolds with a labelled discriminant-$d$ rank-two lattice. The numerical condition (*) is

$$d>6\quad\text{and}\quad d\text{ is not divisible by }4,9\text{, or by a prime }p\equiv 2\pmod{3}.$$

Rationality conjecture. A smooth cubic fourfold $X$ is rational if and only if $X\in\mathcal{C}_d$ for some $d$ satisfying (*).

This is the rationality criterion suggested by the relationship between special cubic fourfolds and associated K3 surfaces. The source gives no resolution status for this assertion.