15.31 (2002)

M. R. Vaughan-Lee and J. Wiegold (Proc. R. Soc. Edinburgh Sect. A, 95 (1983), 215–221) proved that if a finite $p$-group $G$ is generated by elements of breadth $\leqslant n$ (that is, having at most $p^n$ conjugates), then $G$ is nilpotent of class $\leqslant n^2 + 1$; the bound for the class was later improved by A. Mann (J. Group Theory, 4, no. 3 (2001), 241–246) to $\leqslant n^2 - n + 1$. Is there a linear bound for the class of $G$ in terms of $n$?