# Truncated first moment of random variables and stochastic order

Let $$X, Y$$ be two non-negative random variables without first moment, i.e. $$\mathbb{E}[X] = \infty = \mathbb{E}[Y]$$. Suppose for all $$t > 0$$ sufficiently large we have the following inequality for truncated first moments:

$$\mathbb{E}\left[X 1_{\{X \le t \}}\right] \le \mathbb{E}\left[Y 1_{\{Y \le t \}}\right].$$

My question is: is it true that there exists $$C \ge 1$$ such that $$\mathbb{P}(X > t) \le C \mathbb{P}(Y > t)$$ for all $$t$$ sufficiently large?

(I would expect this to hold for $$C = 1$$ but I do not have a clue.)