Determine the dimensional constant a(n) in the Weyl law for the volume spectrum for n ≥ 2

Determine the precise value of the dimensional constant a(n), for dimensions n ≥ 2, appearing in the Weyl law for the volume spectrum of a compact Riemannian manifold (Liokumovich–Marques–Neves), namely the constant in the asymptotic relation that the limit of ω_k^{n+1}/k equals a(n)·vol(M)^{n+1} as k→∞.

Background

The paper recalls the Weyl law for the volume spectrum proved by Liokumovich, Marques, and Neves, which states that the min-max widths ω_k of a compact Riemannian manifold have a universal asymptotic governed by a dimensional constant a(n).

For surfaces (n=1), Chodosh and Mantoulidis determined the exact value a(1)=√π by describing the volume spectrum of the round two-sphere. For higher dimensions, the precise value of a(n) remains undetermined, and the authors explicitly note this unresolved status.