Castravet's conjecture on the rational quotient of elliptic quintics

Let $Y$ be a cubic fourfold, and let ${\mathcal C}$ be the connected component of the Hilbert scheme $\emph{Hilb}^{5m}(Y)$ containing elliptic quintic curves in $Y$. Let the maximally rationally connected quotient of ${\mathcal C}$ be the birational quotient parametrizing its maximally rationally connected fibers. The twisted intermediate Jacobian of $Y$ is the corresponding twisted intermediate Jacobian variety.

Castravet's conjecture. The maximally rationally connected quotient of ${\mathcal C}$ is birationally equivalent to the twisted intermediate Jacobian of $Y$.

The conjecture relates the birational geometry of the Hilbert-scheme component of elliptic quintics to the intermediate Jacobian. The source presents it as an application of the constructed hyperkähler compactification; no resolution is stated.