Uniform volume-doubling failure for domains with vanishing angles

Let $U$ be a bounded domain in Euclidean space or in a Riemannian manifold, and let $\varphi_U$ denote its first Dirichlet eigenfunction. A family of domains satisfies $(\varphi_U^2\textup{-VD})$ uniformly if the associated weighted measures satisfy a uniform volume-doubling property. Vanishing-angle volume-doubling conjecture. Any family of bounded domains $U$ cannot satisfy $(\varphi_U^2\textup{-VD})$ uniformly whenever those domains include at least one angle tending to $0$. This conjecture is motivated by the explicit spherical-wedge examples and by the analogous failure of volume doubling for circular sectors in $\mathbb{R}^2$; the general assertion for families of domains in Euclidean spaces or Riemannian manifolds remains open.