Let $X$ be a random variables with continuou cdf $F$ in the support $[a,b]$. Let $Y$ be another r.v. with cdf $G$, which is continuous in $[a,b)$ and has an atom at $b$. $X$ and $Y$ are independent. We also have $E[X]=E[Y]$ and second order stochastic dominance
$$\int_a^k F(t)\,dt \leq \int_a^k G(t)\,dt, \ \ \ a \leq k \leq b.$$
Can we say the following:
$$\Pr[X \leq Y] \leq \frac 1 2 \text{?}$$
My attempt: I have tried numerical analysis with different parameter values and the above statement is always true. However, it gets difficult to prove it analytically.