Burdzy–Pitman conjecture on the maximal spread of independent coherent distributions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. A random vector $(X,Y)$ is coherent if there exist sub-$\sigma$-fields $\mathcal{G},\mathcal{H}\subset\mathcal{F}$ and an event $A\in\mathcal{F}$ such that

$$X=\mathbb{E}(\mathbbm{1}_A\mid\mathcal{G}),\qquad Y=\mathbb{E}(\mathbbm{1}_A\mid\mathcal{H}).$$

Let $\mathcal{C}$ denote the class of coherent random vectors, and define

$$\mathcal{C}_{\mathcal{I}}=\{(X,Y):X,Y\in\mathcal{C},\ X\perp Y\}.$$

Burdzy–Pitman conjecture. For every $\delta\in(\frac{1}{2},1]$,

$$\sup_{(X,Y)\in\mathcal{C}_{\mathcal{I}}}\mathbb{P}(|X-Y|\geq\delta)=2\delta(1-\delta).$$

Burdzy and Pitman posed this as the sharp bound for the probability that independent coherent variables differ by at least a given threshold. The paper states that its combinatorial and probabilistic results confirm the conjecture, so the conjecture is solved.