A conic bundle over $\mathbb{P}^n$ is a morphism $\pi:X\rightarrow\mathbb{P}^n$ with fibers isomorphic to plane conics. A conic bundle $\pi:X\rightarrow\mathbb{P}^n$ is minimal if it has relative Picard number $1$ and does not have rational sections. The discriminant $D_X$ of $\pi:X\rightarrow\mathbb{P}^n$ is the hypersurface in $\mathbb{P}^n$ parametrizing singular fibers of $\pi:X\rightarrow\mathbb{P}^n$.
Does there exist a (coarse) moduli space for minimal conic bundles over $\mathbb{P}^n$ (modulo birational equivalence) with discriminant of a fixed degree? If so is such a moduli space irreducible?