We generalize several of the main results towards the 1/3-2/3 Conjecture to this new setting: we establish our conjecture when *C* is a weak order interval below a fully commutative element in any acyclic Coxeter group (a generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce *generalized semiorders* for which we resolve the conjecture.

We hope this new perspective may shed light on the proper level of generality in which to consider the 1/3-2/3 Conjecture, and therefore on which methods are likely to be successful in resolving it.

See arXiv version for a full version of this work.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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