Independent coherent opinions bound for radically different opinions

Let $(X,Y)$ be a coherent random vector with values in $[0,1]^2$, and suppose that $X$ and $Y$ are independent. Independent-opinions bound. Then

$$P(|X-Y|\ge 1-\delta)\le 2\delta(1-\delta)$$

for every $\delta\in[0,\frac{1}{2})$. This is a proposed check of a broader claim about coherent pairs; the source does not establish the inequality, and its status is left open.