Optimality conjecture for the dyadic grand coupling of Brownian motions

Let $\{\{\tilde{X}_{\alpha,t}\}_{t}\}_{\alpha\in\mathbb{R}}$ be a grand coupling of Brownian motions $\{\mathrm{BM}(\alpha)\}_{\alpha\in\mathbb{R}}$. A failure probability bound is a function $h$ that bounds the probability that a pair of coupled Brownian motions has not coupled by the relevant time or scale, and an attainable bound is one arising from such a grand coupling. Optimality conjecture. If $h$ is a failure probability bound, then for any $\psi>0$,

$$h(\psi)\ge\int_{\psi/2}^{\infty}\left(\sum_{k=1}^{\infty}2(-1)^{k+1}\exp\left(-\frac{k^{2}\pi^{2}}{2\zeta^{2}}\right)\right)\frac{\zeta^{-2}\psi}{\ln2}\left(\min\left\{ \zeta\psi^{-1},\,1\right\} -\frac{1}{2}\right)\mathrm{d}\zeta,$$

i.e., the attainable failure probability bound given by the dyadic grand coupling is pointwise optimal. The conjecture says that the dyadic grand coupling is globally optimal among grand couplings, although pairwise exactly maximal couplings do not exist; the paper establishes near-maximality and leaves this optimality question open.