Existence of the dihedral family for every number of vertices

Establish that, for every integer $N\ge9$, the dihedral system \eqref{dh:eq} has a solution satisfying $1<c_1<c_2<\cdots<c_M<2$, $T>0$, and the positivity conditions \eqref{dh:pos}.

Background

The dihedral construction reduces the regularity problem for an (N−1)(N-1)-simplex to a finite system in cyclic parameters. The paper certifies solutions for 9≤N≤2019\le N\le201, yielding reflection-free examples in dimensions 8 through 200.

The authors conjecture that the same type of solution exists for every N≥9N\ge9. Such a result would produce reflection-free Morley simplices in every remaining dimension and would imply the broader no-reflections conjecture.

References

The computations suggest the following conjecture. For every $N\ge9$ the system dh:eq has a solution with $1<c_1<c_2<\dots<c_M\<2$ and $T\>0$ which satisfies dh:pos.

dh:eq:

$E_m=4T\sum_{k\inZ/N}r_kr_{k-m}-\frac{4T}{N}\sigma^2-(c_m)=0,\; m=0,1,\dots,M, $

dh:pos:

$h_k=\sum_{j\inZ/N}(r_j)\cos\frac{2\pi jk}{N}>0,\; k=1,\dots,M. $

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 6, Conjecture 6.1 (labelled dh:conj)

Existence of such a profile and convergence of the discrete solutions remain open.

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 6, paragraph following Table 6.1

Does the dihedral system admit other nonregular branches? Can one construct a seven-simplex with regular Morley simplex whose only nonidentity isometry fixes two vertices and interchanges three pairs?

— Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples  (2610.01216 - Tran, 1 Oct 2026) in Section 8, item 3, “Open problems”