Cancellativity of interval monoids for proper quasi-Coxeter elements in F4 and H4

Determine whether the interval monoid M_w is left and right cancellative, and whether the natural homomorphism M_w \to G_w is injective, for every proper quasi-Coxeter element w of type F_4 or H_4 whose interval [1,w] is not a lattice.

Background

The paper answers negatively a question of Baumeister, Holt, Neaime, and Rees for proper quasi-Coxeter elements in types D_n (n at least 5), E_6, E_7, and E_8 by constructing noncancellative interval monoids. In contrast, the authors report that the interval monoid is cancellative in type D_4, even when the associated interval is not a lattice.

The remaining cases are proper quasi-Coxeter elements of types F_4 and H_4 for which [1,w] is not a lattice. The summary table marks cancellativity as unknown for F_4(a_1) and H_4(a_2) through H_4(a_{10}); consequently, injectivity of M_w \to G_w is also unresolved in these cases.

References

The question remains open for the proper quasi-Coxeter elements of types $F_4$ and $H_4$ whose interval $[1,w]$ is not a lattice (see \Cref{tab:exceptional-summary}).

— Noncrossing combinatorics of quasi-Coxeter elements  (2610.11728 - Cainelli et al., 8 Oct 2026) in Introduction, final paragraph before Section 2; Table 1 in Section 5; Section 7