Bott vanishing for Fano 3-folds

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that $H^j(X,{\mathrm{\Omega }}_X^i\otimes L)=0$ for every $j>0$, $i\ge 0$, and $L$ ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are many more than expected. Along the way, we conjecture that for every projective birational morphism $\pi \text{\hspace{0.17em}⁣}:X\to Y$ of smooth varieties, and every line bundle $A$ on $X$ that is ample over $Y$, the higher direct image sheaf $R^j{\pi }_{\ast }({\mathrm{\Omega }}_X^i\otimes A)$ is zero for every $j>0$ and $i\ge 0$.