Quasiconvexity from weak (1,∞)-Poincaré inequality in Euclidean domains

Determine whether every Euclidean domain Ω ⊂ R^n that supports a weak (1,∞)-Poincaré inequality with respect to the Lebesgue measure restricted to Ω is quasiconvex.

Background

This question arises after establishing that, under a doubling assumption, a weak (1,∞)-Poincaré inequality implies (very) ∞-thick quasiconvexity. The authors wonder whether the doubling assumption can be removed when working in Euclidean domains with the Lebesgue measure, and formalize this uncertainty into a precise question.

A positive answer would provide a self-improvement path from local analytical control (weak (1,∞)-Poincaré inequality) to geometric connectivity (quasiconvexity) in domains without assuming a doubling property for the restricted measure.