Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements

Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for oriented-poset lattices. False: the order dimension of the poset of regions can exceed the rank - the Coxeter arrangements $H_4$ and $E_6$ satisfy $\dim W(H_4) \ge 5$ and $\dim W(E_6) \ge 7$.