Secondary-term product conjecture for permutation groups

Let $G_i\subset S_{n_i}$ for $i=1,2$, and let $\operatorname{ind}(G_2)$ denote the relevant index of $G_2$ in its permutation representation. Suppose that the asymptotic distribution of $G_1$-extensions has a secondary term of order $X^{c_1}$, where

$$\operatorname{ind}(G_2)>\frac{n_2}{n_1c_1}.$$

Secondary-term product conjecture. Then the asymptotic distribution of $G_1\times G_2$-extensions has a secondary term of order $X^{c_1/n_2}$. The conjecture proposes a general mechanism behind the secondary terms exhibited for $S_3\times A$-extensions; its resolution is not given in the source.