19.94 (2018)

By definition two elements $g_1$ and $g_2$ of a free metabelian group $F$ with basis $\{x_1, \dots, x_n\}$ have the same distribution if the equations $g_1(x_1, \dots, x_n) = g$ and $g_2(x_1, \dots, x_n) = g$ have the same number of solutions in any finite metabelian group $G$ for every $g \in G$. Is it true that elements $g_1$ and $g_2$ have the same distribution if and only if they are conjugate by some automorphism of $F$?