Let $G$ be a connected semisimple algebraic group over $\mathbb C$, $B_0$ a Borel subgroup, $X$ the flag variety of $G$, $\mathbb O$ a $B_0$-orbit on $X$.
Can we always find a Borel subgroup $B_x$ so that the open $B_x$-orbit $\mathbb O'$ contains $\mathbb O$?
Any help would be appreciated.