It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map to a conic bundle $\pi:Bl_l(X) \to \mathbb{P}^2$ with the same property.
Question: do all 3-fold conic bundle have $H^3(X,\mathcal{O}_X)=0$?