Two-reflection conjecture in dimensions four and five

Prove that every 4- or 5-simplex whose Morley simplex is regular has two hyperplane reflection symmetries interchanging disjoint pairs of vertices.

Background

The paper defines (Gn)(G_n) as the assertion that every nn-simplex with regular Morley simplex has two reflection symmetries interchanging disjoint pairs of vertices. Numerical searches in dimensions four and five found only solutions with this symmetry, motivating the conjecture.

The conjecture remains unresolved in both dimensions. In dimension four, the paper additionally asks whether the regular simplex and the two constructed nonregular examples U1U_1 and U2U_2 exhaust all similarity classes.

References

We conjecture that in dimensions four and five a regular Morley simplex forces two hyperplane reflections interchanging disjoint pairs of vertices.

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 3, Conjecture 3.1 (labelled ld:two-reflections); discussed in Section 1, Results

We conjecture that every dimension $n\ge6$ admits a simplex with regular Morley simplex and no hyperplane reflection symmetry (Conjecture~\ref{hd:all-dimensions}).

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 3, Conjecture 3.2 (labelled hd:all-dimensions); Section 8, item 2

Is there a simplex with regular Morley simplex and trivial isometry group, or one with all edge lengths distinct?

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 8, item 4, “Open problems”