Reverse separation-tree estimate for general Bernoulli index sets

Prove that for every index set T contained in \(\ell^2\), the expected supremum of the Bernoulli process \(\mathcal E_t=\sum_{i\ge1}t_i\varepsilon_i\) satisfies \(\mathsf E\sup_{t\in T}\mathcal E_t\lesssim_r\rho_{\varphi}(T)\), where \(\rho_{\varphi}(T)\) is the parameterized separation-tree functional associated with the Bernoulli distance family \(\varphi_j(s,t)=\sum_{i\ge1}(((r^j(s_i-t_i))^2\wedge1))\) and \(d_0(s,t)=\|s-t\|_2\).

Background

For Bernoulli processes, the paper defines a scale-indexed family of distances from the coordinate differences and uses the abstract separation-tree functional $\rho_\varphi$ to obtain the one-sided estimate $$\rho_\varphi(T)\lesssim_r\mathsf E\sup_{t\in T}\mathcal E_t$$ for arbitrary index sets $T\subset\ell^2$. The unresolved direction is the reverse comparison, which would identify the Bernoulli expected supremum with the separation-tree quantity up to constants.

The paper proves this reverse estimate for bounded, convex, unconditional subsets of $\mathbb R^d$, but does not establish it for arbitrary Bernoulli index sets. The conjectured inequality is therefore a genuine extension beyond the finite-dimensional unconditional setting treated in Section 6.