  Let $\pi:Z\rightarrow\mathbb{P}^{n-1}$ be a general minimal $n$-fold conic
bundle with a hypersurface $B_Z\subset\mathbb{P}^{n-1}$ of degree $d$ as
discriminant. We prove that if $d\geq 4n+1$ then $-K_Z$ is not
pseudo-effective, and that if $d = 4n$ then none of the integral multiples of
$-K_{Z}$ is effective. Finally, we provide examples of smooth unirational
$n$-fold conic bundles $\pi:Z\rightarrow\mathbb{P}^{n-1}$ with discriminant of
arbitrarily high degree.
