Rigidity criterion for standard conic bundles over the projective plane

Let $\pi\colon X\to\mathbb{P}^2$ be a standard conic bundle. Say that it satisfies condition $(\ast)$ when

$$-K_X\notin\operatorname{Int}\operatorname{\overline{NM}}^1(X).$$

Standard conic bundle rigidity conjecture. If $X\to\mathbb{P}^2$ satisfies condition $(\ast)$, then it is birationally rigid. This is proposed as a criterion for birational rigidity of standard conic bundles over the projective plane.