Hyperplane arrangements whose regions all have the same shape

Question

Suppose I have a (finite, real, central, essential) hyperplane arrangement $mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal transformation taking $R$ to $R'$ (these transformations are not required to do anything nice to the rest of the arrangement).  Is $mathcal{H}$ necessarily a reflection arrangement?

Christian Stump · Answer

This is a known open problem (for isometric regions), which, as far as I know, is still not settled.
The dimension 3 case was proved affirmatively in https://arxiv.org/abs/1501.05991, where also some history of the question is outlined. I am not aware of any progress since.

Hyperplane arrangements whose regions all have the same shape

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