Shokurov's rationality criterion for standard conic bundles

Let $S$ be a rational surface, let $\Delta\subset S$ be a discriminant curve, and let $\pi:X\to S$ be a standard conic bundle. Write $K_S$ for the canonical divisor of $S$, $p_{\operatorname{a}}(\Delta)$ for the arithmetic genus of $\Delta$, and $\mathrm{J}_{\mathrm{G}}(X)$ for the Griffiths component of the intermediate Jacobian $\mathrm{J}(X)$. Shokurov's conjecture. The variety $X$ is rational if and only if

$$|2K_S+\Delta|=\varnothing$$

and, when $p_{\operatorname{a}}(\Delta)=6$, the Griffiths component $\mathrm{J}_{\mathrm{G}}(X)$ is trivial.

This is a rationality criterion for three-dimensional standard conic bundles, motivated by the corresponding criterion for surface conic bundles. The condition on the Griffiths component cannot be omitted in the arithmetic-genus-six case, but the general criterion remains conjectural.