Conditioning and covariance on caterpillars

Let X
₁
, ..., X
_n
be joint {±1}-valued random variables. It is known that conditioning on a random subset of O(1/ε
²
) of them reduces their average pairwise covariance to below ε (in expectation). We conjecture that O(1/ε
²
) can be improved to O(1/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
, ..., 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).