Categorical rationality conjecture for fibrations in intersections of two four-dimensional quadrics

Let $X \to \mathbb{P}^1$ be a fibration in complete intersections of two four-dimensional quadrics over an algebraically closed field of characteristic zero. Write $\mathcal{A}_X$ for the residual component of the derived category associated with this fibration. Categorical rationality conjecture. The weak assertion is that the fourfold $X$ is rational if and only if it is categorically representable in codimension at least two. The strong assertion is that $X$ is rational if and only if $\mathcal{A}_X$ is representable in dimension at most two. The rationality of such fourfolds is generally unknown, and these assertions relate birational rationality to categorical representability in the spirit of Kuznetsov's conjecture for cubic fourfolds.