Let $C > c > 0$ and $K > 1$ be constants. Does there exist, for all small enough $\varepsilon > 0$ depending on $c, C, K$, some bound of the following form?

For all random variables $X$ with $c \leq \mathbb E[X^2] \leq C$ and $E[|X|] \leq \varepsilon c^{\frac{1}{2}}$, we have $$\mathbb P\left (|X| \geq K \mathbb E[|X|] \right ) \geq \delta \mathbb E[|X|]^2$$ for some constant $\delta > 0$ depending only on $c, C, K, \varepsilon$.

*Upon some googling, it would seem if we allow $K < 1$, then this is the Paley Zygmund inequality. I wonder if it can be improved to arbitrarily large $K$ in this scenario.*

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