Tyurin's birationality conjecture for Fano threefolds of degree 10

Let $X_{10}$ be a general Fano threefold of degree $10$, and let a quartic double solid be a smooth double cover of $\mathbb{P}^3$ branched over a quartic surface. Tyurin's conjecture. The general Fano threefold $X_{10}$ is birational to a quartic double solid. This conjecture concerns the expected birational analogy between $X_{10}$ and quartic double solids, which have intermediate Jacobians of the same dimension. The source does not provide a resolution of the conjecture.