High-degree discriminant rigidity conjecture for standard conic bundles

Let $\pi\colon X\to\mathbb{P}^2$ be a standard conic bundle, and let its discriminant curve have degree $d$. High-degree discriminant rigidity conjecture. If $d\geq 9$, then $X\to\mathbb{P}^2$ is birationally rigid. The source describes this as a conjecture for which there is little evidence.