Global quadratic upper bound for cubefree group numbers

Prove that the number of isomorphism types of groups of every cubefree order n satisfies $(n)<n^2$, and determine whether the stronger asymptotic relation $(n)=o(n^2)$ also holds.

Background

The paper recalls a conjecture attributed to Blackburn et al. asserting a quadratic upper bound for the number of groups of cubefree order. The authors establish only the asymptotically weaker estimate (n)=n2+o(1)(n)=n^{2+o(1)} and show that the exponent 2 is attained up to subpolynomial factors for infinitely many cubefree integers. They also prove a substantially stronger subpolynomial bound for almost all cubefree orders, but these results do not establish the conjectured global inequality (n)<n2(n)<n^2 or the stronger global statement (n)=o(n2)(n)=o(n^2).

References

For cubefree order $n$, the survey in Section 24.1 shows that $(n)<n8$, and it is conjectured in Conjecture 21.16 that $(n)<n2$ and $(n)=o(n2)$.

— The number of groups of cubefree order  (2608.18815 - Dietrich et al., 19 Aug 2026) in Section 1, Introduction; Section 5, Asymptotic bounds: Theorems A, B, and C