Birational Fano varieties conjecture for cubic fourfolds

Let $X$ and $X'$ be smooth cubic fourfolds, and let $F(X)$ and $F(X')$ be their Fano varieties of lines. Birational Fano varieties conjecture. If

$$F(X)\text{ and }F(X')\text{ are birationally equivalent},$$

then $X$ and $X'$ are birationally equivalent. Equivalently, birational equivalence of the Fano varieties of lines implies birational equivalence of the cubic fourfolds. The conjecture is motivated by examples in which non-isomorphic cubic fourfolds have birationally equivalent Fano varieties of lines, but no general proof is known.