Congruences of conics in $\mathbf{P}^3$

Let $\pi\colon V\to S$ be a conic bundle with a discriminant locus $C\subset S$. We claim the following rationality criterion. $V$ is rational if there exists a pencil of rational curves $\{L_\lambda\subset S|_\lambda\in \mathbf{P}^1\}$ such that either $(L_\lambda\cdot C)\le3$ or $S=\mathbf{P}^3$, $\deg C=5$ and $\pi$ corresponds to an even $\theta$-characteristic. Here we prove the “only if” part of the criterion. The “if” part is reduced to a question of birational equivalence of the congruence of rational curves in $\mathbf{P}^3$ to the congruence of conies.