Suppose $X$ is a pdf over $[0,m]$ and $Y$ is a binary experiment on $X$ such that $P(Y=1|X)$ is continuous, and we have that $\mathbb{E}[X|Y=1] = \mu_y$ and $\mathbb{E}[X] < \mu_y$. Is it always the case that if $\mathbb{E}[X|X>k] = \mu_y$, $\mathbb{E}[X|X\le k] \le \mathbb{E}[X|Y=0]$? This seems intuitive, but is turning out to be deceptively difficult (for me) to prove.

Any ideas for how to prove this simple result, or intuition/counter examples explaining why it is not true, would be very helpful!