Reading–Segovia question on order dimension of posets of regions

The Reading–Segovia question asks whether the order dimension of every simplicial region poset equals the arrangement's rank. Reading established this for most irreducible finite Coxeter types; the exceptional cases remained unresolved, while Segovia posed an analogous question for orientation lattices.

Known results

• Types $A$, $B$, and $D$: dimension equals rank (Reading and collaborators, 2003).
• Supersolvable arrangements: dimension equals rank; type $A$ was previously proved by Flath (1993).
• Earlier exceptional-type bounds included $6\leq\dim(E_6)\leq 9$ and $4\leq\dim(H_4)\leq 6$.

August 2026 counterexamples

A preprint by Daria Poliakova claims explicit counterexamples: $\dim W(H_4)\geq 5>4$ and $\dim W(E_6)\geq 7>6$, with computed values $\dim W(H_4)=5$ and $\dim W(E_6)=8$. It also reports $10\leq\dim W(E_7)\leq 11$ and says the obstruction subgraphs were found by ChatGPT 5.6 Sol Ultra. The claims are not independently verified.

Current status (as of August 2026): The proposed equality is claimed to be disproved by explicit $H_4$ and $E_6$ examples, but the preprint remains unverified and general order dimension is open.