Shokurov's irrationality conjecture for standard conic bundles

Let $\nu\colon V\to U$ be a standard conic bundle, where $V$ is a smooth threefold and $U$ is a smooth rational surface, and let $\Delta$ be its nodal degeneration curve. The notation $|2K_U+\Delta|$ denotes the complete linear system associated with the divisor $2K_U+\Delta$.

Shokurov's conjecture. If

$$|2K_U+\Delta|\ne\varnothing,$$

then $V$ is irrational.

This is an obstruction to rationality for standard conic bundles over rational surfaces. The source attributes the conjecture to Shokurov; the supplied text gives no resolution, so its status remains open.