Suppose there are two CDF's $F$ and $G$ over the same support, $[0,1]$, and assume that one first order stochastically dominates the other: $$\tag{FOSD 1}F\succsim_{FOSD}G$$ meaning that $F(x)\leq G(x),~\forall x\in[0,1]$.
Can we say this fact implies the dominance relation of the conditional distributions of the following? $$\tag{FOSD 2}F(x|x\geq y)\succsim_{FOSD} G(x|x\geq y),~\forall y\in[0,1]$$
Clearly, (FOSD $2$) implies (FOSD $1$) as we can just take $y=0$. However, can we guarantee that the converse also hold? or do we need something more to guarantee (FOSD $2$)?
It is not true. Look at Example 1.3.1. in "Comparison Methods for Stochastic Models and Risks", Alfred Müller, Dietrich Stoyan, 2002.