Conditioning and covariance on caterpillars

Let $X_1,\dots ,X_n$ be joint $\{\pm 1\}$-valued random variables. It is known that conditioning on a random subset of $O(1\mathrm{/}{ϵ}^2)$ of them reduces their average pairwise covariance to below $ϵ$ (in expectation). We conjecture that $O(1\mathrm{/}{ϵ}^2)$ can be improved to $O(1\mathrm{/}ϵ)$. The motivation for the problem and our conjectured improvement comes from the theory of global correlation rounding for convex relaxation hierarchies. We suggest attempting the conjecture in the case that $X_1,\dots ,X_n$ are the leaves of an information flow tree. We prove the conjecture in the case that the information flow tree is a caterpillar graph (similar to a two-state hidden Markov model).