Sharp asymptotic bounds for the eventual-alternation threshold

Determine whether the threshold \Gr(ell), defined as a positive integer such that (-1)^{n-1}G_n^{(ell)}>0 for all n>=\Gr(ell), satisfies 0.47\cdot 3^\ell<\Gr(ell)<0.477\cdot 3^\ell for every ell>=5.

Background

For each order ell, the paper proves that the higher-order Gregory coefficients G_n{(ell)} are eventually alternating and defines \Gr(ell) as an eventual-alternation threshold. The authors compute exact values for several small orders and derive increasingly effective upper bounds using contour-integral estimates.

Numerical computations suggest that \Gr(ell) grows approximately like 3ell. The stated conjecture proposes explicit lower and upper constants for this growth, but no proof is provided.

References

In particular, we would like to know if the following conjecture always holds.

— Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers  (2609.11072 - Xu et al., 10 Sep 2026) in Conjecture following Table 2, Section “Bounds on eventual positiveness of higher order Gregory coefficients”