Motivic rationality conjecture for general special cubic fourfolds

Let $\mathcal{C}_d$ be the special cubic-fourfold divisor whose discriminant $d$ satisfies

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

For a general cubic fourfold $X\in\mathcal{C}_d$, say that its motive is associated to the motive of a K3 surface $S$ if there is an isomorphism

$$t_2(S)(1)\simeq t(X),$$

inducing a Hodge isometry between $T(S)_{\mathbf{Q}}(1)$ and $T(X)_{\mathbf{Q}}$. Motivic rationality conjecture. The cubic fourfold $X$ is rational if and only if its motive is associated to the motive of a K3 surface. This is proposed as a motivic reformulation relating rationality to the transcendental motive; the source gives no resolution.